New Infinite Families of Perfect Quaternion Sequences and Williamson Sequences

TL;DR

This paper introduces new infinite families of perfect quaternion sequences of lengths 2^t, disproves a conjecture about their maximum length, and establishes the existence of a broad class of Williamson sequences linked to combinatorial design theory.

Contribution

It provides the first constructions of perfect quaternion sequences for all lengths 2^t and links Williamson sequences to combinatorial design theory, expanding known sequence families.

Findings

Perfect quaternion sequences exist for all lengths 2^t.

Disproves Mow's conjecture on maximum sequence length.

Establishes infinite classes of Williamson sequences for lengths multiple of powers of two.

Abstract

We present new constructions for perfect and odd perfect sequences over the quaternion group $Q_8$. In particular, we show for the first time that perfect and odd perfect quaternion sequences exist in all lengths $2^t$ for $t\geq0$. In doing so we disprove the quaternionic form of Mow's conjecture that the longest perfect $Q_8$-sequence that can be constructed from an orthogonal array construction is of length 64. Furthermore, we use a connection to combinatorial design theory to prove the existence of a new infinite class of Williamson sequences, showing that Williamson sequences of length $2^t n$ exist for all $t\geq0$ when Williamson sequences of odd length $n$ exist. Our constructions explain the abundance of Williamson sequences in lengths that are multiples of a large power of two.

Equations

$$i^2=j^2=k^2=-1,ij=k,jk=i,ki=j.$$

$$C_{A,B}(t):=\sum _{r=0}^{n-t-1}{a}_r{b}_{r+t}^{\ast }$$

$$R_{A,B}(t)$$

$${\hat{R}}_{A,B}(t)$$

$$R_{A,B}(t):=\sum _{r=0}^{n-1}{a}_r{b}_{r+t\mathrm{mod}n}^{\ast }$$

$${\hat{R}}_{A,B}(t):=\sum _{r=0}^{n-1}(-1{)}^{\lfloor (r+t)/n\rfloor }{a}_r{b}_{r+t\mathrm{mod}n}^{\ast }.$$

$$\sum _{r=0}^{n-1}\sum _{s=0}^{m-1}{a}_{r,s}{a}_{r+t\mathrm{mod}n,s+t^{\prime }\mathrm{mod}n}^{\ast }=0$$

$$\left[\begin{array}{cc}\text{+}& \text{+}\\ \text{+}& \text{-}\end{array}\right]$$

$$\begin{array}{ccccccccc}{a}_r& \text{-}& \text{+}& \text{+}& \text{+}& \text{+}& \text{+}& \text{+}& \text{+}\\ b_r& \text{-}& \text{-}& \text{+}& \text{-}& \text{-}& \text{+}& \text{+}& \text{-}\\ c_r& \text{-}& \text{-}& \text{-}& \text{+}& \text{-}& \text{-}& \text{+}& \text{+}\\ d_r& \text{-}& \text{+}& \text{-}& \text{-}& \text{-}& \text{+}& \text{-}& \text{+}\\ & & & & & & & & \\ s_r& 1& i& j& k& q& qi& qj& qk\end{array}$$

$$(d(A)\text{x}n(A^{\prime }),d(B)\text{x}n(B^{\prime }),d(C)\text{x}n(C^{\prime }),d(D)\text{x}n(D^{\prime }))$$

$$R_{A_{\text{new}}}(t)+R_{B_{\text{new}}}(t)+R_{C_{\text{new}}}(t)+R_{D_{\text{new}}}(t)=0\text{ for 1≤t<4n}.$$

$$\begin{cases}R_{\operatorname{d}(X)}(t/2)+R_{\operatorname{n}(X^{\prime})}(t/2)&\text{if $t$ is even,}\\ R_{\operatorname{d}(X),\operatorname{n}(X^{\prime})}\bigl{(}\frac{t-1}{2}\bigr{)}+R_{\operatorname{n}(X^{\prime}),\operatorname{d}(X)}\bigl{(}\frac{t+1}{2}\bigr{)}&\text{if $t$ is odd}.\end{cases}$$

$$R_{X_{\text{new}}}(t)=2R_X(t/2)+2{\hat{R}}_{X^{\prime }}(t/2).$$

$$\sum _{X=A,B,C,D}{R}_X(t/2)=0\text{and}\sum _{X=A,B,C,D}{\hat{R}}_{X^{\prime }}(t/2)=0$$

$$\mathrm{sum}(A^{\prime }{)}^2+\mathrm{sum}(B^{\prime }{)}^2+\mathrm{sum}(C^{\prime }{)}^2+\mathrm{sum}(D^{\prime }{)}^2=4n.$$

$$(n(A^{\prime })\text{x}d(A),n(B^{\prime })\text{x}d(B),n(C^{\prime })\text{x}d(C),n(D^{\prime })\text{x}d(D))$$

$$([A;B;\tilde{B};\tilde{A}],[\tilde{B};\tilde{A};A;B],[-\tilde{B};\tilde{A};A;-B],[-A;B;\tilde{B};-\tilde{A}])$$

$$([A;B;\tilde{B};\tilde{A};A;B;\tilde{B};\tilde{A}],$$

$$[A;B;-\tilde{B};-\tilde{A};-A;-B;\tilde{B};\tilde{A}],$$

$$[A;-B;\tilde{B};-\tilde{A};-A;B;-\tilde{B};\tilde{A}],$$

$$[A;-B;-\tilde{B};\tilde{A};A;-B;-\tilde{B};\tilde{A}])$$

$$\left[\begin{array}{cccc}\text{+}& \text{+}& \text{+}& \text{+}\\ \text{+}& \text{+}& \text{-}& \text{-}\\ \text{+}& \text{-}& \text{+}& \text{-}\\ \text{+}& \text{-}& \text{-}& \text{+}\end{array}\right]$$

$$\begin{array}{c}([A;B;\tilde{B};\tilde{A}],[A;B;-\tilde{B};-\tilde{A}],\\ [A;-B;\tilde{B};-\tilde{A}],[A;-B;-\tilde{B};\tilde{A}])\end{array}$$

$$([A;B;\tilde{B};\tilde{A};-A;-B;-\tilde{B};-\tilde{A}],$$

$$[A;B;-\tilde{B};-\tilde{A};-A;-B;\tilde{B};\tilde{A}],$$

$$[A;-B;\tilde{B};-\tilde{A};-A;B;-\tilde{B};\tilde{A}],$$

$$[A;-B;-\tilde{B};\tilde{A};-A;B;\tilde{B};-\tilde{A}])$$

$$\begin{array}{r}(-1{)}^{\lfloor (nm-r-1)/n\rfloor +\lfloor (nm-r-1)/m\rfloor }{x}_{r\mathrm{mod}n}{y}_{r\mathrm{mod}m}\\ =(-1{)}^{n+m+\lfloor -(r+1)/n\rfloor +\lfloor -(r+1)/m\rfloor }{x}_{r\mathrm{mod}n}{y}_{r\mathrm{mod}m}.\end{array}$$

$$(-1{)}^{1+\lfloor r/n\rfloor +\lfloor r/m\rfloor }{x}_{r\mathrm{mod}n}{y}_{r\mathrm{mod}m}$$

$$P:=[-A;jB;k\tilde{B};i\tilde{A};iA;kB;j\tilde{B};-\tilde{A}]$$

$${\hat{R}}_P(t)=C_{[P^{\prime };{\tilde{P}}^{\prime }]}(t)-C_{[P^{\prime };{\tilde{P}}^{\prime }]}(8n-t{)}^{\ast }$$

$$=C_{P^{\prime }}(t)+C_{{\tilde{P}}^{\prime }}(t)+C_{{\tilde{P}}^{\prime },P^{\prime }}(4n-t{)}^{\ast }-C_{P^{\prime },{\tilde{P}}^{\prime }}(4n-t{)}^{\ast }.$$

$$\varphi (t)$$

$$\psi (t)$$

$$\psi (t)=-j{C}_{B,A}(t)+i{C}_{\tilde{B},B}(t)+j{C}_{B,A}(t)=i{C}_{\tilde{B},B}(t).$$

Full text

††publicationid: pubid:

0018-9448 © 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.

See https://www.ieee.org/publications/rights/index.html for more information.

I Introduction

Sequences that have zero correlation with themselves after a nontrivial cyclic shift are known as perfect [1]. Such sequences have a long history [2] and an amazing wealth of applications, for example, appearing in the 3GPP LTE standard [3]. Perfect sequences and their generalizations have been applied to spread spectrum multiple access systems [4], radar systems [5], fast start-up equalization [6], channel estimation and synchronization [7], peak-to-average power ratio reduction [8], image watermarking [9], and constructing complementary sets [10]. They also tend to have small aperiodic correlation [11] and small ambiguity function sidelobes [12].

One of the first researchers to study perfect sequences was Heimiller [13] who in 1961 gave a construction for perfect sequences using matrices with array orthogonality. His construction generated perfect sequences of length $p^{2}$ over the complex $p$th roots of unity for any prime $p$. Shortly after Heimiller’s paper was published, Frank and Zadoff published a response [14] pointing out that Frank had discovered the same construction a decade prior as an aircraft engineer at the Sperry Gyroscope Company. Moreover, Frank was granted a patent for a communication system based on his sequences [15].

Frank’s construction generates perfect sequences of length $n^{2}$ over the complex $n$th roots of unity. Given the theoretical elegance of perfect sequences and their importance to many fields of engineering it would be extremely interesting and useful if a construction could be found that produces perfect sequences longer than $n^{2}$ using $n$th roots of unity. However, over 60 years of effort has failed to find such a construction and it has been conjectured by Mow [16] that such a construction does not exist. In light of this, several researchers have searched for perfect sequences over other alphabets such as the group of quaternions [17]. Quaternions are generalizations of the complex numbers that include the additional numbers $j$ and $k$ that satisfy the relationships

[TABLE]

Note that these relationships imply that quaternions are generally noncommutative.

Communication systems have been described that are based on the quaternions [18, 19] and standard mathematical techniques like the Fourier transform have been generalized to the quaternions for usage in signal processing [20] and image processing [21]. Perfect sequences over the quaternions were first studied by Kuznetsov [22]. Kuznetsov and Hall [23] showed that Mow’s conjecture cannot be directly extended to these sequences by constructing a perfect sequence over a quaternion alphabet with $24$ elements and whose length is over 5 billion. In 2012, Barrera Acevedo and Hall [24] constructed perfect sequences over the alphabet $\{\pm 1,\pm i,j\}$ in all lengths of the form $q+1$ where $q\equiv 1\pmod{4}$ is a prime power. Most recently, Blake [25] has run extensive searches for perfect sequences over the basic quaternion alphabet $Q_{8}\coloneqq\{\pm 1,\pm i,\pm j,\pm k\}$ (see Section VI for more on previous work). The longest perfect $Q_{8}$-sequence generated from an orthogonal array construction that Blake found had length $64$. Intriguingly, this length is exactly the square of the alphabet size of $8$. This led Blake to conjecture a quaternionic form of Mow’s conjecture that the quadratic Frank–Heimiller bound applies to perfect sequences over $Q_{8}$ generated from an orthogonal array construction.

In this paper we show that Blake’s conjecture is false. This is accomplished via a new construction for perfect quaternion sequences that can be used to construct arbitrarily long perfect $Q_{8}$-sequences (see Section V) and odd perfect $Q_{8}$-sequences (see Section IV). In particular, we construct the first infinite family of odd perfect quaternion sequences and show for the first time that perfect and odd perfect quaternion sequences of length $2^{t}$ exist for all $t$. This is starkly different from what happens if one restricts the alphabet to only include the purely real or complex elements of $Q_{8}$. It is known that the longest perfect $\{\pm 1\}$-sequence of length $2^{t}$ is of length four [26] and the longest perfect $\{\pm 1,\pm i\}$-sequence of length $2^{t}$ is of length sixteen [27]. Additionally, the longest odd perfect $\{\pm 1\}$-sequence has length two [28].

Recently Barrera Acevedo and Dietrich [29, 30] discovered a correspondence between perfect sequences over the quaternions and Williamson sequences from combinatorial design theory [31] (see Section II for the definition of Williamson sequences). Using this relationship we construct new Williamson sequences in even lengths, including all powers of two. Prior to our construction Williamson sequences of length $2^{t}$ were only known to exist for $t\leq 6$ [32]. See Section III for our Williamson sequence construction and Section IV for a construction for a variant of Williamson sequences that our Williamson sequence construction relies on.

In 1944, when Williamson introduced the sequences that now bear his name [31] he showed that the existence of Williamson sequences of odd length $n$ implied the existence of Williamson sequences of length $2n$. In 1970, twenty-six years later, Turyn [27] generalized Williamson’s result by showing that the existence of Williamson sequences of odd length $n$ implied the existence of Williamson sequences of lengths $2^{t}n$ for $t\leq 4$. Nearly fifty years after Turyn’s result, Barrera Acevedo and Dietrich [29] improved this to $t\leq 6$ in 2019. In this paper we complete this process of generalizing Williamson’s doubling result by showing the result in fact holds for all $t\geq 0$. In other words, the existence of Williamson sequences of odd length $n$ implies the existence of Williamson sequences of length $2^{t}n$ for all $t$ and this provides a large new infinite class of Williamson sequences.

Exhaustive searches for Williamson sequences [33, 34] have shown that they exist in all lengths $n<65$ except for $n=35$, $47$, $53$, and $59$. They are generally very abundant in the even lengths (particularly in the lengths that are divisible by a large power of two) but not in the odd lengths. For example, over 50,000 inequivalent sets of Williamson sequences exist in length $64$ but fewer than 100 sets of Williamson sequences exist in all odd lengths up to $65$. Previously this dichotomy was unexplained but the constructions that we provide in this paper produce approximately 75% of the Williamson sequences that exist in all even lengths $n\leq 70$ (see Section VII).

II Preliminaries

In this section we provide the preliminaries necessary to explain our construction for perfect quaternion sequences, odd perfect quaternion sequences, and Williamson sequences. First we define the concept of sequence perfection in terms of the amount of correlation that a sequence has with cyclically shifted copies of itself. Following this, we discuss the array orthogonality property used in several constructions for perfect sequences. We then define Williamson sequences and finally present Barrera Acevedo and Dietrich’s equivalence between perfect quaternion sequences and Williamson sequences.

II-A Complementary sequences and perfect sequences

The aperiodic crosscorrelation of two sequences $A=[a_{0},\dotsc,a_{n-1}]$ and $B=[b_{0},\dotsc,b_{n-1}]$ of length $n$ is given by

[TABLE]

where $z^{*}$ denotes the conjugate of $z$ [35]. For a general quaternionic number we have $(a+bi+cj+dk)^{*}\coloneqq a-bi-cj-dk$ where $a$, $b$, $c$, $d$ are real. The aperiodic autocorrelation of $A$ is given by $C_{A}(t)\coloneqq C_{A,A}(t)$. More generally, we also define the periodic and odd periodic (or negaperiodic) crosscorrelations by

[TABLE]

The periodic and odd periodic autocorrelations of $A$ are given by $R_{A}(t)\coloneqq R_{A,A}(t)$ and $\hat{R}_{A}(t)\coloneqq\hat{R}_{A,A}(t)$ respectively. These functions may be extended to all integers $t$ via the expressions

[TABLE]

A set $S$ of sequences of length $n$ is called complementary if $\sum_{A\in S}C_{A}(t)=0$ for all $1\leq t<n$.111Technically $S$ should be defined to be a multiset but we follow standard convention and refer to it as a set. Similarly, $S$ is called periodic complementary if $\sum_{A\in S}R_{A}(t)=0$ and odd periodic complementary (or negacomplementary) if $\sum_{A\in S}\hat{R}_{A}(t)=0$ for all $1\leq t<n$. A single sequence $A$ is called perfect if $R_{A}(t)=0$ for all $1\leq t<n$ and odd perfect if $\hat{R}_{A}(t)=0$ for all $1\leq t<n$.

Note that the periodic correlation values $R_{A}(t)$ of a sequence are preserved under the cyclic shift operator $[a_{0},\dotsc,a_{n-2},a_{n-1}]\mapsto[a_{n-1},a_{0},\dotsc,a_{n-2}]$ and the negaperiodic correlation values $\hat{R}_{A}(t)$ of a sequence are preserved under the negacyclic shift operator $[a_{0},\dotsc,a_{n-2},a_{n-1}]\mapsto[-a_{n-1},a_{0},\dotsc,a_{n-2}]$ (see [36, 37]). Therefore applying a cyclic shift to any sequence in a set of periodic complementary sequences or applying a negacyclic shift to any sequence in a set of negacomplementary sequences does not disturb the property of the set being periodic complementary or negacomplementary.

Let $(-1)*X$ denote the sequence whose $r$th entry is $(-1)^{r}x_{r}$, i.e., the alternating negation operation. A set of periodic complementary sequences of odd length $n$ can be converted into a set of negacomplementary sequences and vice versa by applying the alternating negation operation (see [38]).

Lemma 1**.**

If $n$ is odd then $(A,B,C,D)$ are periodic complementary sequences of length $n$ if and only if $((-1)*A,(-1)*B,(-1)*C,(-1)*D)$ are negacomplementary sequences.

Example 1**.**

$(\verb|+--|,\verb|+--|,\verb|+--|,\verb|+++|)$ is a set of complementary sequences of length $3$ and $(\verb|++-|,\verb|++-|,\verb|++-|,\verb|+-+|)$ is the set of negacomplementary sequences generated from it using Lemma 1. Note that we follow the convention of writing $1$s by $\verb|+|$ and $-1$s by $\verb|-|$.

II-B Matrices with array orthogonality

The sequences in a set $S$ are periodically uncorrelated if any two distinct sequences $A$, $B$ in $S$ satisfy $R_{A,B}(t)=0$ for all $0\leq t<n$. If a set of periodic complementary sequences $S=\{S_{1},\dotsc,S_{m}\}$ (with each sequence of length $n$ a multiple of $m$) are periodically uncorrelated then the $n\times m$ matrix whose columns are given by $S_{1}$, $\dotsc$, $S_{m}$ is said to have array orthogonality.

Matrices with array orthogonality are important because they are used in many constructions for perfect sequences such as in the Frank–Heimiller and Mow constructions. In particular, the sequence formed by concatenating the rows of a matrix with array orthogonality is perfect. Additionally, matrices with array orthogonality are themselves perfect arrays. Perfect arrays are often studied as a way of generalizing the concept of perfection from sequences to matrices (e.g., see [39, 40]). They are defined to be $n\times m$ matrices $A=(a_{r,s})$ that satisfy

[TABLE]

for all $(t,t^{\prime})\neq(0,0)$ with $0\leq t<n$, $0\leq t^{\prime}<m$.

Example 2**.**

The columns of the matrix

[TABLE]

have array orthogonality and therefore this is a perfect array. It generates the perfect sequence $[\verb|+++-|]$.

II-C Williamson and nega Williamson sequences

First, we describe the symmetry properties that are used in the definition of Williamson sequences and will be important in our construction for Williamson sequences. A sequence $A$ of length $n$ is called symmetric if $a_{t}=a_{n-t}$ for all $1\leq t<n$ and is palindromic if $a_{t}=a_{n-t-1}$ for all $0\leq t<n$. For example, $[x,y,z,y]$ is symmetric and $[y,z,y]$ is palindromic. Note that a sequence is symmetric if and only if the subsequence formed by removing its first element is palindromic. Additionally, we call a sequence antipalindromic if $a_{t}=-a_{n-t-1}$ for all $0\leq t<(n-1)/2$ and antisymmetric if $a_{t}=-a_{n-t}$ for all $1\leq t<n/2$. If $[X;Y]$ denotes sequence concatenation and $\tilde{X}$ denotes the reversal of $X$ then $[X;\tilde{X}]$ is palindromic and $[X;-\tilde{X}]$ is antipalindromic.

A quadruple of symmetric $\{\pm 1\}$-sequences $(A,B,C,D)$ are known as Williamson if they are periodic complementary, i.e., if $R_{A}(t)+R_{B}(t)+R_{C}(t)+R_{D}(t)=0$ for all $1\leq t<n$. Additionally, we call a quadruple of $\{\pm 1\}$-sequences $(A,B,C,D)$ nega Williamson if they are negacomplementary, i.e., $\hat{R}_{A}(t)+\hat{R}_{B}(t)+\hat{R}_{C}(t)+\hat{R}_{D}(t)=0$ for all $1\leq t<n$.

As is conventional, we require by definition that Williamson sequences are symmetric. If the symmetry condition is replaced with the weaker property that the sequences are instead amicable (i.e., $R_{X,Y}(t)=R_{Y,X}(t)$ for all $t$ and $X,Y\in\{A,B,C,D\}$) then the sequences are known as Williamson-type. For nega Williamson sequences we do not require them to be symmetric. Instead, our work has discovered the importance of palindromic and antipalindromic nega Williamson sequences; see Section III for details.

Example 3**.**

$(\verb|++|,\verb|++|,\verb|+-|,\verb|+-|)$ are Williamson sequences of length $2$. Similarly, $(\verb|+-|,\verb|+-|,\verb|+-|,\verb|+-|)$ are antipalindromic nega Williamson sequences and $(\verb|++|,\verb|++|,\verb|++|,\verb|++|)$ are palindromic nega Williamson sequences. $(\verb|++-+|,\verb|++-+|,\verb|++-+|,\verb|++-+|)$ are Williamson sequences of length $4$. Similarly, $(\verb|+-+-|,\verb|+-+-|,\verb|++--|,\verb|++--|)$ are antipalindromic nega Williamson sequences and $(\verb|+--+|,\verb|+--+|,\verb|++++|,\verb|++++|)$ are palindromic nega Williamson sequences.

II-D The Barrera Acevedo–Dietrich correspondence

Let $Q_{8}\coloneqq\{\pm 1,\pm i,\pm j,\pm k\}$ be the quaternion group and $Q_{+}\coloneqq Q_{8}\cup qQ_{8}$ where $q\coloneqq(1+i+j+k)/2$ (note that $Q_{+}$ is a set of sixteen quaternions that is not a group). Barrera Acevedo and Dietrich [29, Theorem 2.4] show that there is an equivalence between Williamson-type sequences and perfect sequences over $Q_{+}$.

Theorem 1**.**

There is a one-to-one correspondence between sets of Williamson-type sequences of length $n$ and perfect sequences of length $n$ over $Q_{+}$.

This correspondence is made explicit through the following mapping between the $r$th entries of a set of Williamson-type sequences $(a_{r},b_{r},c_{r},d_{r})$ and the $r$th entry of the corresponding perfect sequence $s_{r}$:

[TABLE]

Additionally, we have the rule that if $(a_{r},b_{r},c_{r},d_{r})$ maps to $s_{r}$ then $(-a_{r},-b_{r},-c_{r},-d_{r})$ maps to $-s_{r}$.

Because of this theorem a construction for Williamson sequences of length $n$ also produces symmetric perfect $Q_{+}$-sequences. In this paper we state our construction in terms of Williamson sequences with the understanding that it equivalently produces perfect $Q_{+}$-sequences. In Section V we also show how our construction can be used to produce perfect $Q_{8}$-sequences in many lengths including all powers of two.

If the entries of a set of Williamson sequences $(A,B,C,D)$ of length $n$ satisfy $a_{r}b_{r}c_{r}d_{r}=1$ for all $0\leq r<n$ then the Barrera Acevedo–Dietrich correspondence produces perfect $Q_{8}$-sequences. For this reason we say that a quadruple of sequences has the $Q_{8}$-property if the entries of its sequences satisfy $a_{r}b_{r}c_{r}d_{r}=1$ for all $0\leq r<n$. In his original paper [31] Williamson proved that the entries of all Williamson sequences in odd lengths $n$ satisfy $a_{r}b_{r}c_{r}d_{r}=-a_{0}b_{0}c_{0}d_{0}$ for $1\leq r<n$. As a consequence, no Williamson sequence of odd length $n>1$ can have the $Q_{8}$-property. However, many Williamson sequences in even lengths have the $Q_{8}$-property. Exhaustive searches [34] have found Williamson sequences with the $Q_{8}$-property in all even lengths $n\leq 2^{5}$ except for $6$, $12$, and $28$.

Barrera Acevedo and Dietrich also use the periodic product construction of Lüke [41] that generates perfect sequences from shorter perfect sequences. Let $X\times Y$ be the sequence whose $r$th entry is $x_{r\bmod n}y_{r\bmod m}$ for $0\leq r<nm$ (where $X$ has length $n$ and $Y$ has length $m$).

Theorem 2**.**

Suppose $X$ and $Y$ have coprime lengths $n$ and $m$. If $X$ is a perfect $Q_{8}$-sequence and $Y$ is a perfect $Q_{+}$-sequence then $X\times Y$ is a perfect $Q_{+}$-sequence. Furthermore, if $X$ and $Y$ are symmetric then $X\times Y$ is symmetric.

Theorems 1 and 2 immediately yield the following corollary.

Corollary 1**.**

If a perfect symmetric $Q_{8}$-sequence of length $2^{t}$ exists and Williamson sequences of odd length $n$ exist then Williamson sequences of length $2^{t}n$ exist.

Barrera Acevedo and Dietrich also provide examples of symmetric perfect $Q_{8}$-sequences for all $t\leq 6$. In this paper we extend their result by showing that perfect symmetric $Q_{8}$-sequences of length $2^{t}$ exist for all $t\geq 0$ and therefore show Williamson sequences exist in all lengths of the form $2^{t}n$ whenever Williamson sequences exist in odd length $n$.

Example 4**.**

Using the Barrera Acevedo–Dietrich correspondence the Williamson sequences $(\verb|++-+|,\verb|++-+|,\verb|++-+|,\verb|++-+|)$ produce the perfect $Q_{8}$-sequence $[\verb|--+-|]$, the Williamson sequences $(\verb|++--+|,\verb|-+--+|,\verb|-++++|,\verb|-++++|)$ produce the perfect $Q_{+}$-sequence $[\verb|q-jj-|]$ and the Williamson sequences $(\verb|++--+--+|,\verb|++--+--+|,\verb|+++-+-++|,\verb|+++-+-++|)$ produce the perfect $Q_{8}$-sequence $[\verb|--j+-+j-|]$. (We denote $i$, $j$, $k$, and $q$ by $\verb|i|$, $\verb|j|$, $\verb|k|$, and $\verb|q|$.)

III Constructions for Williamson sequences

Our main construction is based on the following simple sequence operations.

The doubling of $X$, denoted by $\operatorname{d}(X)$, i.e., $\operatorname{d}(X)\coloneqq[x_{0},\dotsc,x_{n-1},x_{0},\dotsc,x_{n-1}]$. 2. 2.

The negadoubling of $X$, denoted by $\operatorname{n}(X)$, i.e., $\operatorname{n}(X)\coloneqq[x_{0},\dotsc,x_{n-1},-x_{0},\dotsc,-x_{n-1}]$. 3. 3.

The interleaving of $X$ and $Y$, denoted by $X\mathbin{\text{x}}Y$, i.e., $X\mathbin{\text{x}}Y\coloneqq[x_{0},y_{0},x_{1},y_{1},\dotsc,x_{n-1},y_{n-1}]$.

We will use the following properties of these operations in our construction. For completeness, proofs of these properties are given in the appendix. In each property $X$ and $Y$ are arbitrary sequences of the same length and $t$ is an arbitrary integer.

$R_{\operatorname{d}(X)}(t)=2R_{X}(t)$. 2. 2.

$R_{\operatorname{n}(X)}(t)=2\hat{R}_{X}(t)$. 3. 3.

$R_{\operatorname{d}(X),\operatorname{n}(Y)}(t)=0$ and $R_{\operatorname{n}(Y),\operatorname{d}(X)}(t)=0$. 4. 4.

$R_{X\mathbin{\text{x}}Y}(2t)=R_{X}(t)+R_{Y}(t)$. 5. 5.

$R_{X\mathbin{\text{x}}Y}(2t+1)=R_{X,Y}(t)+R_{Y,X}(t+1)$. 6. 6.

If $X$ is symmetric then $\operatorname{d}(X)$ is symmetric. 7. 7.

If $X$ is antipalindromic and of even length then $\operatorname{n}(X)$ is palindromic. 8. 8.

If $X$ is symmetric and $Y$ is palindromic then $X\mathbin{\text{x}}Y$ is symmetric.

Our main construction for Williamson sequences is given by the following theorem.

Theorem 3**.**

If $(A,B,C,D)$ are Williamson sequences of even length $n$ and $(A^{\prime},B^{\prime},C^{\prime},D^{\prime})$ are antipalindromic nega Williamson sequences of length $n$ then

[TABLE]

are Williamson sequences of length $4n$.

This theorem can be used to construct a large number of new Williamson sequences, assuming that sequences that satisfy the preconditions are known. For example, if $N$ Williamson sequences of length $n$ are known and $M$ antipalindromic nega Williamson sequences of length $n$ are known this theorem immediately implies that at least $NM$ Williamson sequences of length $4n$ exist.

We now prove Theorem 3.

Proof.

Let $X_{\text{new}}\coloneqq\operatorname{d}(X)\mathbin{\text{x}}\operatorname{n}(X^{\prime})$ for $X\in\{A,B,C,D\}$. By construction it is clear that $X_{\text{new}}$ is a $\{\pm 1\}$-sequence of length $4n$. Additionally, $X_{\text{new}}$ is symmetric by property 8 since $\operatorname{d}(X)$ is symmetric by property 6 and $\operatorname{n}(X^{\prime})$ is palindromic by property 7. It remains to show that

[TABLE]

Note that by properties 4 and 5 we have $R_{X_{\text{new}}}(t)$ is

[TABLE]

By property 3 we have $R_{X_{\text{new}}}(t)=0$ when $t$ is odd. When $t$ is even by properties 1 and 2 we have

[TABLE]

When $t=2n$ we have $R_{X}(t/2)=n$ and $\hat{R}_{X^{\prime}}(t/2)=-n$ (because $X$ and $X^{\prime}$ both have length $n$) so $R_{X_{\text{new}}}(t)=0$ in this case. Otherwise we have

[TABLE]

for even $t\neq 2n$ with $1\leq t<4n$ since $(A,B,C,D)$ are Williamson sequences and $(A^{\prime},B^{\prime},C^{\prime},D^{\prime})$ are nega Williamson sequences. It follows that $\sum_{X=A,B,C,D}R_{X_{\text{new}}}(t)=0$, as required. ∎

Example 5**.**

Using the set of Williamson sequences $(\verb|++-+|,\verb|++-+|,\verb|++-+|,\verb|++-+|)$ and the set of antipalindromic nega Williamson sequences $(\verb|+-+-|,\verb|+-+-|,\verb|++--|,\verb|++--|)$ in Theorem 3 produces the set of Williamson sequences

$\mathllap{(}\verb|+++--++-+-++--++|,\verb|+++--++-+-++--++|\mathrlap{,}$

$\verb|++++--+-+-+--+++|,\verb|++++--+-+-+--+++|\mathrlap{).}$

Note that the assumption that $n$ is even is essential to the theorem. If $n$ is odd the constructed sequences will not be symmetric. Additionally, antipalindromic nega Williamson sequences do not exist in odd lengths $n>1$, as we now show.

Lemma 2**.**

Antipalindromic nega Williamson sequences do not exist in odd lengths except for $n=1$.

Proof.

Let $(A,B,C,D)$ be a hypothetical set of antipalindromic nega Williamson sequences of odd length $n$. Note that a sequence $X$ of odd length $n$ is antipalindromic if and only if $X^{\prime}\coloneqq(-1)*X$ is antipalindromic. By Lemma 1, it follows that $(A^{\prime},B^{\prime},C^{\prime},D^{\prime})$ are antipalindromic periodic complementary sequences. If $\operatorname{sum}(X)$ denotes the row sum of $X$ it is well-known (e.g., via the Wiener–Khinchin theorem or sequence compression [42]) that

[TABLE]

However, if $X^{\prime}$ is antipalindromic we have $\operatorname{sum}(X^{\prime})=\pm 1$ and the above sum of squared row sums must be equal to four, implying that $n=1$. ∎

Although the construction in Theorem 3 requires sequences of even lengths there is a variant of the construction that uses sequences of odd lengths. The proof is similar and uses the following additional properties.

If $X$ is antisymmetric and of odd length then $\operatorname{n}(X)$ is symmetric. 2. 10)

If $X$ is palindromic then $\operatorname{d}(X)$ is palindromic.

Theorem 4**.**

Let $n$ be odd and let $(A,B,C,D)$ and $(A^{\prime},B^{\prime},C^{\prime},D^{\prime})$ be quadruples of sequences of length $n$. If $(A,B,C,D)$ is the result of applying $(n-1)/2$ cyclic shifts to each member in a set of Williamson sequences and $(A^{\prime},B^{\prime},C^{\prime},D^{\prime})$ is the result of applying $(n+1)/2$ negacyclic shifts to each member in a set of palindromic nega Williamson sequences then

[TABLE]

are Williamson sequences of length $4n$.

Proof.

Since a symmetric sequence of odd length $n$ becomes palindromic after $(n-1)/2$ cyclic shifts, $(A,B,C,D)$ are palindromic. Since a palindromic sequence of odd length $n$ becomes antisymmetric after $(n+1)/2$ negacyclic shifts, $(A^{\prime},B^{\prime},C^{\prime},D^{\prime})$ are antisymmetric.

Let $X_{\text{new}}\coloneqq\operatorname{n}(X^{\prime})\mathbin{\text{x}}\operatorname{d}(X)$ for $X\in\{A,B,C,D\}$. By construction it is clear that $X_{\text{new}}$ is a $\{\pm 1\}$-sequence of length $4n$. Additionally, $X_{\text{new}}$ is symmetric by property 8 since $\operatorname{n}(X^{\prime})$ is symmetric by property 9 and $\operatorname{d}(X)$ is palindromic by property 10. The remainder of the proof now proceeds as in the proof of Theorem 3. ∎

Example 6**.**

Note $(\verb|++--+|,\verb|-+--+|,\verb|-++++|,\verb|-++++|)$ are Williamson sequences and $(\verb|+---+|,\verb|++-++|,\verb|+---+|,\verb|+++++|)$ are palindromic nega Williamson sequences. Then using $(A,B,C,D)=(\verb|-+++-|,\verb|-+-+-|,\verb|++-++|,\verb|++-++|)$ and $(A^{\prime},B^{\prime},C^{\prime},D^{\prime})=(\verb|++-+-|,\verb|+--++|,\verb|++-+-|,\verb|---++|)$ in Theorem 4 produces the following Williamson sequences of length $20$:

$\mathllap{(}\verb|+-++-+++-----+++-++-|,\verb|+--+--+++---+++--+--|\mathrlap{,}$

$\verb|++++--++-+-+-++--+++|,\verb|-+-+--+++++++++--+-+|\mathrlap{).}$

IV Nega Williamson sequences and odd perfect sequences

Although antipalindromic nega Williamson sequences do not exist in odd lengths larger than $1$ we now present constructions showing that they exist in many even lengths. First, note that when the length is even there is a correspondence between palindromic and antipalindromic nega Williamson sequences. In other words, to use Theorem 3 it is sufficient to find palindromic nega Williamson sequences.

Lemma 3**.**

There is a one-to-one correspondence between palindromic and antipalindromic nega Williamson sequences in even lengths.

Proof.

Applying $n/2$ negacyclic shifts to each sequence in a set of palindromic nega Williamson sequences of even length $n$ yields a set of antipalindromic nega Williamson sequences. Similarly, a set of palindromic nega Williamson sequences can be produced from a set of antipalindromic nega Williamson sequences by applying the negacyclic shift operator $n/2$ times. ∎

Example 7**.**

$(\verb|+--++-|,\verb|+-+-+-|,\verb|++-+--|,\verb|+++---|)$ are antipalindromic nega Williamson sequences that may be converted into the palindromic nega Williamson sequences $(\verb|--++--|,\verb|+-++-+|,\verb|-++++-|,\verb|++++++|)$ using the translation in Lemma 3.

We now provide a construction that shows that infinitely many palindromic nega Williamson sequences exist. Recall that $\tilde{X}$ denotes the reverse of the sequence $X$ and $[X;Y]$ denotes the concatenation of $X$ and $Y$. Note that if $X$ and $Y$ have length $n$ then $C_{[X;Y]}(t)=C_{X}(t)+C_{Y}(t)+C_{Y,X}(n-t)^{*}$ and $C_{[X;Y]}(2n-t)=C_{X,Y}(n-t)$ for $0\leq t\leq n$. First we prove a simple lemma.

Lemma 4**.**

Applying the negadoubling operator to the sequences in a complementary set produces a negacomplementary set.

Proof.

Note that if $X$ is a sequence of length $n$ then $\hat{R}_{\operatorname{n}(X)}(t)=2C_{X}(t)$ for $0\leq t\leq n$; by definition we have $\hat{R}_{\operatorname{n}(X)}(t)=C_{\operatorname{n}(X)}(t)-C_{\operatorname{n}(X)}(2n-t)^{*}$ and when $0\leq t\leq n$ we have $C_{\operatorname{n}(X)}(t)=C_{X}(t)+C_{-X}(t)+C_{-X,X}(n-t)^{*}=2C_{X}(t)-C_{X}(n-t)^{*}$ and $C_{\operatorname{n}(X)}(2n-t)=C_{X,-X}(n-t)=-C_{X}(n-t)$. Thus $\hat{R}_{\operatorname{n}(X)}=2C_{X}(t)-C_{X}(n-t)^{*}+C_{X}(n-t)^{*}=2C_{X}(t)$.

Suppose $S$ is a complementary set of sequences of length $n$. Using the above property we have $\sum_{A\in S}\hat{R}_{\operatorname{n}(A)}(t)=2\sum_{A\in S}C_{A}(t)$ for all $0\leq t\leq n$. Since $S$ is complementary this implies $\sum_{A\in S}\hat{R}_{\operatorname{n}(A)}(t)=0$ for all $1\leq t\leq n$. Using the symmetry $\hat{R}_{X}(t)=\hat{R}_{X}(-t)^{*}$ shows that this also holds for all $n<t<2n$. ∎

Example 8**.**

Using the set of four complementary sequences $(\verb|+++|,\verb|+--|,\verb|+-+|,\verb|++-|)$ of length three with Lemma 4 produces the set of four negacomplementary sequences $(\verb|+++---|,\verb|+---++|,\verb|+-+-+-|,\verb|++---+|)$ of length six.

Theorem 5**.**

If $A$ and $B$ are complementary $\{\pm 1\}$-sequences (i.e., $(A,B)$ is a Golay pair) then

[TABLE]

and

[TABLE]

are sets of palindromic nega Williamson sequences with the $Q_{8}$-property.

Proof.

The fact that the sequences are palindromic is immediate from the manner in which they were constructed. Note that $(\tilde{A},\tilde{B})$ is a Golay pair since $(A,B)$ is a Golay pair. Thus $(A,B,\tilde{B},\tilde{A})$ is a complementary set and since

[TABLE]

is an orthogonal matrix the sequences

[TABLE]

form a complementary set of sequences by [43, Thm. 7]. Since they are complementary they are also negacomplementary and applying $2n$ negacyclic shifts (where $A$ and $B$ are of length $n$) to the second and third sequences and negating the fourth shows the first set in the theorem is negacomplementary.

Since (IV) are complementary, by Lemma 4 applying the negadoubling operator to these sequences will produce negacomplementary sequences. In other words,

[TABLE]

are negacomplementary. Applying $4n$ negacyclic shifts (where $A$ and $B$ are of length $n$) to the first and last sequences shows the second set in the theorem is negacomplementary.

Lastly, we show that the produced sequences $X$, $Y$, $U$, and $V$ have the $Q_{8}$-property. In the first set we have $x_{r}=a_{r}$, $y_{r}=b_{n-r+1}$, $u_{r}=-y_{r}$, and $v_{r}=-x_{r}$ for all $0\leq r<n$ and in the second set we have $x_{r}=y_{r}=u_{r}=v_{r}=a_{r}$ for all $0\leq r<n$. In each case $x_{r}y_{r}u_{r}v_{r}=1$ for all $0\leq r<n$ and more generally we have $x_{r}y_{r}u_{r}v_{r}=1$ for all $0\leq r<4n$ in the first set (and $0\leq r<8n$ in the second set). ∎

Example 9**.**

Using the Golay pair $(\verb|++|,\verb|+-|)$ the first set of palindromic nega Williamson sequences generated by Theorem 5 is $(\verb|+++--+++|,\verb|-++++++-|,\verb|+-++++-+|,\verb|--+--+--|)$; the second set is $(\verb|+++--++++++--+++|,\verb|+++-+------+-+++|,\verb|++-+-+----+-+-++|,\verb|++-++-++++-++-++|)$.

Golay sequences, originally defined by Golay [44], are known to exist in lengths $2$, $10$, and $26$ [45]. Since Turyn has shown that Golay pairs in lengths $n$ and $m$ can be composed to form Golay pairs in length $nm$ [46] they also exist in all lengths of the form $2^{a}5^{b}13^{c}$ with $a\geq b+c$. Thus, Theorem 5 implies that palindromic nega Williamson sequences with the $Q_{8}$-property exist in all lengths of the form $2^{a}5^{b}13^{c}$ with $a\geq b+c+2$. In particular, taking $b=c=0$ gives that palindromic nega Williamson sequences with the $Q_{8}$-property exist in all lengths that are powers of two (since palindromic nega Williamson sequences with the $Q_{8}$-property of length $2$ exist, see Example 3). These are not the only lengths in which palindromic nega Williamson sequences exist, however. Palindromic nega Williamson sequences in many other lengths may be constructed using Lüke’s product construction with odd perfect sequences.

First, note that in odd lengths there is an equivalence between Williamson sequences and palindromic nega Williamson sequences.

Lemma 5**.**

There is a one-to-one correspondence between palindromic nega Williamson sequences and Williamson sequences in odd lengths.

Proof.

Let $(A,B,C,D)$ be a set of Williamson sequences of odd length $n$. By Lemma 1, $((-1)*A,(-1)*B,(-1)*C,(-1)*D)$ will be a set of nega Williamson sequences. The sequences in this set will be antisymmetric since if $X$ is symmetric and of odd length then $(-1)*X$ is antisymmetric. Applying $(n-1)/2$ negacyclic shifts to each sequence in this set produces a set of palindromic nega Williamson sequences. Similarly, arbitrary palindromic nega Williamson sequences of odd length can be transformed into Williamson sequences by applying the inverse of the above transformations. ∎

Example 10**.**

The set of Williamson sequences $(\verb|-++--++|,\verb|---++--|,\verb|-+----+|,\verb|-+----+|)$ generates the set of palindromic nega Williamson sequences $(\verb|++---++|,\verb|--+-+--|,\verb|+-----+|,\verb|+-----+|)$ and vice versa using the transformation in Lemma 5.

Since Williamson sequences are known to exist for all lengths $n<35$ this implies that palindromic nega Williamson sequences exist for all odd lengths up to $33$.

Next, we note that odd perfect sequences in even lengths can be composed with odd perfect sequences in odd lengths to generate longer odd perfect sequences. Let $X\mathbin{\hat{\times}}Y$ be the sequence whose $r$th entry is $(-1)^{\lfloor r/n\rfloor+\lfloor r/m\rfloor}x_{r\bmod n}y_{r\bmod m}$ for $0\leq r<nm$ (where $X$ has length $n$ and $Y$ has length $m$).

Lemma 6**.**

Suppose $X$ and $Y$ have coprime lengths $n$ and $m$, one of which is even. If $X$ is odd perfect and $Y$ is odd perfect then $X\mathbin{\hat{\times}}Y$ is odd perfect. Furthermore, if $X$ and $Y$ are palindromic then $X\mathbin{\hat{\times}}Y$ is antipalindromic.

Proof.

The fact that $X\mathbin{\hat{\times}}Y$ is odd perfect follows from $\hat{R}_{X\mathbin{\hat{\times}}Y}(t)=\hat{R}_{X}(t)\hat{R}_{Y}(t)$ as given in [28, Eq. 9]. Suppose that $0\leq r<nm$ is arbitrary. Since $X$ and $Y$ are palindromic we have $x_{nm-r-1\bmod n}y_{nm-r-1\bmod m}=x_{n-r-1\bmod n}y_{m-r-1\bmod m}=x_{r\bmod n}y_{r\bmod m}$. Thus the $(nm-r-1)$th entry of $X\mathbin{\hat{\times}}Y$ is

[TABLE]

Using the fact that $\lfloor-(r+1)/s\rfloor=-(\lfloor r/s\rfloor+1)$ for positive integers $r$, $s$ and that $n+m$ is odd this becomes

[TABLE]

which is the negative of the $r$th entry of $X\mathbin{\hat{\times}}Y$ as required. ∎

Example 11**.**

The palindromic odd perfect sequences $X=[\verb|++|]$ and $Y=[\verb|+q+|]$ used with Lemma 6 produces the antipalindromic odd perfect sequence $[\verb|+q-+Q-|]$. (We use $\verb|Q|$ to denote $-q$.)

We now show that infinitely many palindromic odd perfect $Q_{8}$-sequences exist using a variant of Theorem 5 and the Barrera Acevedo–Dietrich correspondence.

Theorem 6**.**

If $(A,B)$ is a Golay pair then

[TABLE]

is a palindromic odd perfect $Q_{8}$-sequence.

Proof.

This follows by a direct but tedious calculation of $\hat{R}_{P}(t)$; we give an example of how this may be done. Let $P=[P^{\prime};\tilde{P}^{\prime}]$ have length $8n$. We find for $0\leq t\leq 4n$ that

[TABLE]

Suppose that $1\leq t\leq n$. Then $C_{P^{\prime}}(t)=\phi(t)+\psi(n-t)^{*}$ where

[TABLE]

Since multiplying a sequence by a constant does not change its autocorrelation values and since $(A,B,\tilde{A},\tilde{B})$ are complementary we find that $\phi(t)=0$. Furthermore, using the fact $C_{xA,yB}(t)=xy^{*}C_{A,B}(t)$ and $C_{\tilde{A},\tilde{B}}(t)=C_{B,A}(t)$ (since $A$ and $B$ have real entries) we find

[TABLE]

Additionally, we have $C_{\tilde{P}^{\prime}}(t)=C_{P^{\prime}}(t)^{*}$, $C_{P^{\prime},\tilde{P}^{\prime}}(4n-t)=C_{-A,-\tilde{A}}(n-t)=C_{A,\tilde{A}}(n-t)$, and $C_{\tilde{P}^{\prime},P^{\prime}}(4n-t)=C_{iA,i\tilde{A}}(n-t)=C_{A,\tilde{A}}(n-t)$. Then

[TABLE]

Similarly one can show $\hat{R}_{P}(t)=0$ for $n<t\leq 4n$ from which the symmetry $\hat{R}_{X}(t)=\hat{R}_{X}(-t)^{*}$ implies $\hat{R}_{P}(t)=0$ for all $1\leq t<8n$. ∎

Example 12**.**

Using the Golay pair $(\verb|++|,\verb|+-|)$ with Theorem 6 produces the palindromic odd perfect sequence $[\verb|--jJKkiiiikKJj--|]$. (We denote $-i$ by $\verb|I|$, $-j$ by $\verb|J|$, and $-k$ by $\verb|K|$.)

In particular, palindromic odd perfect sequences exist in all lengths that are a power of two since Theorem 6 implies they exist in all lengths of the form $2^{a}5^{b}13^{c}$ with $a\geq b+c+3$ and they exist in the lengths $2$ and $4$ as shown by the examples $[\verb|++|]$ and $[\verb|+ii+|]$. Using the fact that palindromic nega Williamson sequences exist in all odd lengths up to $33$ we used the Barrera Acevedo–Dietrich correspondence to construct palindromic odd perfect sequences in all odd lengths up to $33$. (This relies on the fact that a set of palindromic real sequences $\{A,B,C,D\}$ are necessarily “nega-amicable” in that $\hat{R}_{X,Y}(t)=\hat{R}_{Y,X}(t)$ for all $t$ and $X,Y\in\{A,B,C,D\}$.) Furthermore, using the fact that palindromic odd perfect $Q_{8}$-sequences exist in all lengths that are powers of two we used Lemma 6 to construct palindromic odd perfect sequences in all even lengths up to $68$ (see the appendix for an explicit list). The Barrera Acevedo–Dietrich correspondence applied to these sequences gives palindromic nega Williamson sequences in all even lengths up to $68$. It is conceivable that palindromic odd perfect sequences and palindromic nega Williamson sequences actually exist in all even lengths.

V Perfect quaternion sequences

We now use our constructions for Williamson sequences and palindromic nega Williamson sequences to show that Williamson sequences and perfect $Q_{8}$-sequences exist in all lengths $2^{t}$ with $t\geq 0$.

Theorem 7**.**

Symmetric perfect sequences over $Q_{8}$ exist for all lengths $2^{t}$.

Proof.

First, note that Golay sequences exist in all lengths $2^{t}$. By Theorem 5 and Lemma 3 it follows that antipalindromic nega Williamson sequences with the $Q_{8}$-property exist in all lengths $2^{t}$ for $t\geq 2$ (they also exist for smaller $t$, see Example 3). Theorem 3 then implies that if Williamson sequences with the $Q_{8}$-property exist in length $2^{t}$ they also exist in length $2^{t+2}$ for all $t\geq 1$. Additionally, it is known that Williamson sequences with the $Q_{8}$-property exist in lengths $2$ and $4$ (see below). By induction, Williamson sequences with the $Q_{8}$-property exist in all lengths $2^{t}$ for $t\geq 0$. By the Barrera Acevedo–Dietrich correspondence symmetric perfect $Q_{8}$-sequences exist in all lengths $2^{t}$ as well. ∎

Example 13**.**

We use the base Golay pair $(\verb|+|,\verb|+|)$, the base Williamson sequences $(\verb|+|,\verb|+|,\verb|+|,\verb|+|)$, $(\verb|+-|,\verb|+-|,\verb|++|,\verb|++|)$, and the base nega Williamson sequences $(\verb|++|,\verb|++|,\verb|++|,\verb|++|)$. Additionally, we use Golay’s interleaving doubling construction [44] to generate larger Golay pairs via the mapping $(A,B)\mapsto(A\mathbin{\text{x}}B,A\mathbin{\text{x}}-B)$. From Theorem 4 we generate the Williamson sequences $(\verb|++-+|,\verb|++-+|,\verb|++-+|,\verb|++-+|)$, from Theorem 5 and Lemma 3 we generate the antipalindromic nega Williamson sequences $(\verb|++++|,\verb|++++|,\verb|-++-|,\verb|-++-|)$, and from Theorem 3 we generate the Williamson sequences $(\verb|+-+++++-|,\verb|+-+++++-|,\verb|+--+++--|,\verb|+--+++--|)$. Continuing in this way and using the Barrera Acevedo–Dietrich correspondence produces perfect $Q_{8}$-sequences of lengths $2^{t}$ for all $t\geq 0$. We denote $-i$ by $\verb|I|$, $-j$ by $\verb|J|$, and $-k$ by $\verb|K|$ and explicitly give the sequences produced with this construction for $t\leq 7$:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The perfect sequences generated by Theorem 7 for $t\geq 7$ are counterexamples to the quaternionic form of Mow’s conjecture presented by Blake [25] because they can be generated using an orthogonal matrix construction, as we now show.

Theorem 8**.**

Let $n\geq 32$ and let $M$ be the $(n/4)\times 4$ matrix containing the entries of a perfect sequence $P$ generated by Theorem 7 using the second set from Theorem 5. Write the entries of $P$ in $M$ from left to right and top to bottom (i.e., $M_{i,j}=P_{4i+j}$). Then the columns of $M$ have the array orthogonality property.

Proof.

Let $M_{0}$, $M_{1}$, $M_{2}$, $M_{3}$ denote the columns of $M$ as sequences. We must show that $\sum_{r=0}^{3}R_{M_{r}}(t)=0$ for all $1\leq t<n/4$ and that $R_{M_{r},M_{s}}(t)=0$ for all $t$ and $0\leq r<s<4$.

By construction $P=(M_{0}\mathbin{\text{x}}M_{2})\mathbin{\text{x}}(M_{1}\mathbin{\text{x}}M_{3})$ is a perfect sequence and therefore

[TABLE]

By property 4 in Section III this implies $R_{M_{0}\mathbin{\text{x}}M_{2}}(t)+R_{M_{1}\mathbin{\text{x}}M_{3}}(t)=0$ for all $1\leq t<n/2$ and $\sum_{r=0}^{3}R_{M_{r}}(t)=0$ for all $1\leq t<n/4$.

Since $P$ was generated by applying the Barrera Acevedo–Dietrich correspondence to the sequences generated in Theorem 3 we have $P=\operatorname{d}(A)\mathbin{\text{x}}\operatorname{n}(B)$ for some perfect symmetric sequence $A$ and antipalindromic sequence $B$ of even length $n/4$. Let $X^{\prime}$ denote the sequence formed by the even entries of $X$ and let $X^{\prime\prime}$ denote the sequence formed by the odd entries of $X$. Then we have

[TABLE]

By property 3 of Section III this representation yields $R_{M_{0},M_{1}}(t)=R_{M_{0},M_{3}}(t)=R_{M_{1},M_{2}}(t)=R_{M_{2},M_{3}}(t)=0$ for all $t$ and only the crosscorrelation of the pairs $(M_{0},M_{2})$ and $(M_{1},M_{3})$ are left to consider. Since $A=A^{\prime}\mathbin{\text{x}}A^{\prime\prime}$ is perfect and was generated by applying Theorem 3 we have that $A^{\prime}$ is of the form $\operatorname{d}(C_{1})$ for some $C_{1}$ and $A^{\prime\prime}$ is of the form $\operatorname{n}(C_{2})$ for some $C_{2}$. Property 3 of Section III then yields $R_{M_{0},M_{2}}(t)=2R_{A^{\prime},A^{\prime\prime}}(t)=2R_{\operatorname{d}(C_{1}),\operatorname{n}(C_{2})}(t)=0$ for all $t$.

Lastly, we must show $R_{M_{1},M_{3}}(t)=0$. Suppose the antipalindromic $B$ was generated from Theorem 5 using the Golay pair $(D,E)$. An analysis of the Barrera Acevedo–Dietrich correspondence shows that we have

[TABLE]

We have $\hat{R}_{B^{\prime},B^{\prime\prime}}(t)=C_{B^{\prime},B^{\prime\prime}}(t)-C_{B^{\prime\prime},B^{\prime}}(n/8-t)^{*}$ by definition. Suppose that $0\leq t\leq n/32$ so that $C_{B^{\prime\prime},B^{\prime}}(n/8-t)=C_{kE,-k\tilde{E}}(n/32-t)=-C_{E,\tilde{E}}(n/32-t)$ and $C_{B^{\prime},B^{\prime\prime}}(t)=\phi(t)+\psi(n/32-t)^{*}$ where

[TABLE]

Note that $C_{\tilde{E},\tilde{D}}(t)=C_{D,E}(t)$ since $D$ and $E$ contain real entries. Then

[TABLE]

Finally, R_{M_{1},M_{3}}(t)=2\hat{R}_{B^{\prime},B^{\prime\prime}}(t)=2\bigl{(}-C_{E,\tilde{E}}(n/32-t)^{*}-(-C_{E,\tilde{E}}(n/32-t))^{*}\bigr{)}=0 for $0\leq t\leq n/32$. A similar calculation shows $\hat{R}_{B^{\prime},B^{\prime\prime}}(t)=0$ for $n/32<t<n/8$ from which it follows that $R_{M_{1},M_{3}}(t)=0$ for all $t$. ∎

Example 14**.**

For $n=16$ we use the perfect sequence found in Example 13 and the matrix it generates has the array orthogonality property as well. Otherwise we give the matrices constructed using Theorem 8 for $n=32$, $64$, and $128$. The transpose of the matrices are displayed to save space.

[TABLE]

VI Previous Work

Despite an enormous amount of work the conjecture that the longest perfect sequences over the complex $n$th roots of unity have length $n^{2}$ remains open, though some special cases of this conjecture have been resolved. In the case $n=2$, Turyn [26] showed that any perfect binary sequence of length longer than four must have length $4m^{2}$ for odd $m$ that is not a prime power. Despite this progress the general conjecture even for $n=2$ remains open [47]. In the case $n=4$, the conjecture states that the longest perfect sequence over $\{\pm 1,\pm i\}$ has length $16$ and Turyn [27] showed that the length of any longer perfect sequence cannot be of the form $2p^{k}$ for any prime $p$ and integer $k$. Most recently, Ma and Ng [48] showed many restrictions on the length of perfect sequences over the $p$th roots of unity for prime $p$. In particular, they showed that no perfect sequences of length $2p^{k+2}$ or $p^{k+3}$ exist for $k\geq 0$.

Several other constructions for perfect sequences over complex roots of unity have been found since the construction of Frank and Heimiller. In 1972, Chu described a method [49] of producing perfect sequences of any length $n$. Shortly after Chu’s paper was published, Frank published a response [50] pointing out that Zadoff had discovered the same construction and had been granted a patent for a communication system based on his construction [51]. Other variants of Zadoff and Chu’s sequences have been described by Alltop [52] and Lewis and Kretschmer [53]. In 1983, Milewski [54] found a new construction for perfect sequences of length $n^{2t+1}$ over $n^{t+1}$th roots of unity for all $n,t\geq 1$. In 2004, Liu and Fan [55] found a new construction for perfect sequences of length $n$ over $n$th roots of unity when $n$ is a multiple of four. In 2014, Blake and Tirkel [56] gave a construction for perfect sequences of length $4mn^{t+1}$ over $2mn^{t}$th roots of unity for $m,n,t\geq 1$.

Blake has run extensive searches for perfect sequences over complex roots of unity and quaternions [25] and has found a number of perfect sequences over the $n$th roots of unity that cannot be generated using matrices with array orthogonality (like those from the Frank–Heimiller and Mow constructions). The sequences that he found are counterexamples to the conjecture that Mow’s unified construction produces all perfect sequences over $n$th roots of unity [16] but are shorter than $n^{2}$ and thus are not counterexamples to Mow’s original conjecture [57] (also generalized by Lüke, Schotten, and Hadinejad-Mahram [28] to odd perfect sequences).

Apparently no infinite family of odd perfect $Q_{8}$-sequences has been previously constructed.222A construction given in [25] uses the term ‘odd-perfect’ but with an alternative meaning that $R_{A}(t)=0$ for odd $t$. It is known that no infinite family of odd perfect sequences can exist over the alphabet $\{\pm 1\}$—Lüke, Schotten, and Hadinejad-Mahram show that the longest odd perfect $\{\pm 1\}$-sequence has length two and they conjecture that the longest odd perfect sequence over the alphabet $\{\pm 1,\pm i\}$ has length four [28]. However, Lüke has constructed almost perfect and odd perfect $\{\pm 1,\pm i\}$-sequences $A$ (with $R_{A}(t)=0$ or $\hat{R}_{A}(t)=0$ for all $1\leq t<n$ except $t=n/2$) in all lengths $q+1$ where $q\equiv 1\pmod{4}$ is a prime power [58].

Some work has also been done on perfect quaternion arrays. In 2013, Barrera Acevedo and Jolly [59] constructed perfect $Q_{8}$-arrays of size $2\times(p+1)/2$ for primes $p\equiv 1\pmod{4}$. Furthermore, they extended a construction of Arasu and de Launey [60] to produce perfect $Q_{8}$-arrays of size $2p\times p(p+1)/2$ for primes $p\equiv 1\pmod{4}$. Additionally, Blake [25] found a construction for perfect $Q_{8}$-arrays of size $2^{t}\times 2^{t}$ with $2\leq t<7$.

There does not seem to be much prior work on nega Williamson sequences, though Xia, Xia, Seberry, and Wu [38] study them under the name “4-suitable negacyclic matrices”. They give constructions for them in terms of Golay pairs, Williamson sequences, and base sequences. However, we could find no prior constructions that specifically generated palindromic or antipalindromic nega Williamson sequences. Lüke presents a method [61] of constructing pairs of negacomplementary binary sequences, also studied under the name “associated pairs” by Ito [62] and “negaperiodic Golay pairs” by Balonin and Đoković [63]. Lüke and Schotten [64] also give a construction for odd perfect almost binary sequences that Wen, Hu, and Jin [65] use to construct negacomplementary binary sequences. Jin et al. [66] give necessary conditions for the existence of negacomplementary sequences and a doubling construction for negacomplementary sequences. Yang, Tang, and Zhou [67] show that a $\{\pm 1\}$-sequence $A$ of length $n$ satisfies \max_{1\leq t<n}\bigl{\lvert}\hat{R}_{A}(t)\bigr{\rvert}-1\geq(n-1)\bmod{2} and give constructions for sequences that are optimal, i.e., ones that meet the bound exactly.

Williamson sequences have been quite well-studied since introduced by Williamson 75 years ago [31]. They are often presented using matrix notation and known as “Williamson matrices” since one of their traditional applications has been to construct Hadamard matrices. In this formulation Williamson matrices are defined to be circulant (i.e., each row is the cyclic shift of the previous row) and Williamson sequences are simply the first rows of Williamson matrices. Baumert and Hall [68] performed an exhaustive search for Williamson sequences of odd length $n\leq 23$ and presented a doubling construction for a generalization of Williamson matrices that are symmetric but not circulant. Turyn [69] constructed Williamson sequences in all lengths $(q+1)/2$ where $q\equiv 1\pmod{4}$ is a prime power, Whiteman [70] constructed Williamson sequences in all lengths $q(q+1)/2$ where $q\equiv 1\pmod{4}$ is a prime power, and Spence [71] constructed Williamson sequences in all lengths $q^{t}(q+1)/2$ where $q\equiv 1\pmod{4}$ is a prime power and $t\geq 0$.

Computer searches have determined that Williamson sequences in odd lengths are particularly rare. Following Baumert and Hall’s exhaustive search, Baumert [72] found a new set of Williamson sequences in length $29$, Sawade [73] found eight new sets in lengths $25$ and $27$, Yamada [74] found one new set in length $37$, Koukouvinos and Kounias [75] found four new sets in length $33$, Đoković [76, 77, 78] found six new sets in the lengths $25$, $31$, $33$, $37$, and $39$, van Vliet found one new set in length $51$ (unpublished but appears in [33]), Holzmann, Kharaghani, and Tayfeh-Rezaie [33] found one new set in length $43$, and Bright, Kotsireas, and Ganesh [34] found one new set in length $63$. Đoković [77] also found that no Williamson sequences exist in length $35$ and Holzmann, Kharaghani, and Tayfeh-Rezaie [33] found that no Williamson sequences exist in lengths $47$, $53$, and $59$.

In even lengths Williamson sequences are much more common, as first shown by an exhaustive search up to length $18$ by Kotsireas and Koukouvinos [79]. Non-exhaustive searches were performed up to length $34$ by Bright et al. [80], and up to length $42$ by Zulkoski et al. [81]. Bright [82] completed an exhaustive search up to length $44$ and Bright, Kotsireas, and Ganesh [34] completed an exhaustive search in the even lengths up to $70$. Barrera Acevedo and Dietrich’s correspondence [29] can be used to generate Williamson sequences in many lengths including $70$.

VII Conclusion

We have shown that perfect and odd perfect $Q_{8}$-sequences exist in all lengths that are a power of two. Barrera Acevedo and Dietrich [30] summarize the knowledge (as of 2018) of perfect $Q_{8}$-sequences as follows:

Currently there exists only one infinite family of perfect sequences over the quaternions (of magnitude one)…

This infinite family was found by Barrera Acevedo and Hall [24] who gave a construction for perfect $Q_{8}$-sequences in lengths of the form $q+1$ where $q\equiv 1\pmod{4}$ is a prime power. Thus, our construction for lengths of the form $2^{t}$ is the second known infinite family of perfect $Q_{8}$-sequences. Additionally, our construction for odd perfect $Q_{8}$-sequences is the first known infinite family of odd perfect $Q_{8}$-sequences. Because our perfect sequences can be constructed using matrices with array orthogonality (as shown in Theorem 8) they disprove Blake’s conjecture [25, Conjecture 8.2.1] that the longest perfect $Q_{8}$-sequences generated from an orthogonal array construction have length $64$. Theorem 8 also implies the existence of perfect $Q_{8}$-arrays of sizes $2^{t}\times 4$ for all $t\geq 2$ and the construction of Barrera Acevedo and Jolly [59] then implies the existence of perfect $Q_{8}$-arrays of size $2^{t}p\times 4p$ when $p=2^{t+2}-1$ is prime.

Furthermore, we generalize Williamson’s doubling construction [31] from 1944, showing that the existence of Williamson sequences of odd length $n$ implies not only the existence of Williamson sequences of length $2n$ but also implies the existence of Williamson sequences of length $2^{t}n$ for all $t\geq 1$. We have also shown the importance of nega Williamson sequences, a class of sequences defined by Xia et al. [38] in 2006. In particular, we have demonstrated the importance of palindromic nega Williamson sequences.

Lastly, our constructions provide an explanation for the abundance of Williamson sequences in lengths that are divisible by a large power of two. Prior to this work it was noticed that Williamson sequences are much more abundant in these lengths. For example, fewer than 100 sets of Williamson sequences are known to exist in the odd lengths, but an exhaustive computer search [34] found 130,739 sets of Williamson sequences in the even lengths up to $70$. We found that it was possible to generate 95,759 (about 75%) of these sets using Theorems 3 and 4. Thus, these theorems provide an explanation for the existence of many Williamson sequences. However, they still do not explain the existence of Williamson sequences in all even lengths. In particular, they only work in lengths that are multiples of four.

It would also be interesting to find a construction that works in the even lengths that are not multiples of four. Williamson’s doubling result can be used for lengths $2n$ but requires that Williamson sequences of odd length $n$ exist. Barrera Acevedo and Dietrich’s correspondence can be used in certain cases even if Williamson sequences of length $n$ do not exist, assuming $n$ is composite. For example, the Barrera Acevedo–Dietrich correspondence implies that Williamson sequences of length $70$ exist since Williamson sequences of length $7$ exist and Williamson sequences with the $Q_{8}$-property of length $10$ exist. However, we could only generate about 40% of the Williamson sequences in length $70$ using the Barrera Acevedo–Dietrich correspondence suggesting that there is another construction for Williamson sequences that is currently unknown.

Appendix

VII-A Proofs

We now give proofs of the properties from Section III. In each case we let $X=[x_{0},\dotsc,x_{n-1}]$ and $Y=[y_{0},\dotsc,y_{n-1}]$ be arbitrary sequences of length $n$.

$R_{\operatorname{d}(X)}(t)=2R_{X}(t)$.

The $r$th entry of $\operatorname{d}(X)$ is $x_{r\bmod n}$. Then

[TABLE] 2. 2.

$R_{\operatorname{n}(X)}(t)=2\hat{R}_{X}(t)$.

The $r$th entry of $\operatorname{n}(X)$ is $(-1)^{\lfloor r/n\rfloor}x_{r\bmod n}$. Then $R_{\operatorname{n}(X)}(t)$ is

[TABLE] 3. 3.

$R_{\operatorname{d}(X),\operatorname{n}(Y)}(t)=0$ and $R_{\operatorname{n}(Y),\operatorname{d}(X)}(t)=0$.

The $r$th entry of $\operatorname{d}(X)$ is $x_{r\bmod n}$ and the $r$th entry of $\operatorname{n}(Y)$ is $(-1)^{\lfloor r/n\rfloor}y_{r\bmod n}$. Then $R_{\operatorname{d}(X),\operatorname{n}(Y)}(t)$ is

[TABLE]

The second property follows because $R_{X,Y}(t)=R_{Y,X}(-t)^{*}$ for all $X$, $Y$, and $t$. 4. 4.

$R_{X\mathbin{\text{x}}Y}(2t)=R_{X}(t)+R_{Y}(t)$.

The $(2r)$th entry of $X\mathbin{\text{x}}Y$ is $x_{r}$ and the $(2r+1)$th entry is $y_{r}$. Then $R_{X\mathbin{\text{x}}Y}(2t)$ is

[TABLE] 5. 5.

$R_{X\mathbin{\text{x}}Y}(2t+1)=R_{X,Y}(t)+R_{Y,X}(t+1)$.

The $(2r)$th entry of $X\mathbin{\text{x}}Y$ is $x_{r}$ and the $(2r+1)$th entry is $y_{r}$. Then $R_{X\mathbin{\text{x}}Y}(2t+1)$ is

[TABLE] 6. 6.

If $X$ is symmetric then $\operatorname{d}(X)$ is symmetric.

Note that $x_{2n-r\bmod n}=x_{n-r\bmod n}=x_{r\bmod n}$. Thus the $(2n-r)$th entry of $\operatorname{d}(X)$ is equal to the $r$th entry, as required. 7. 7.

If $X$ is antipalindromic and of even length then $\operatorname{n}(X)$ is palindromic.

Let $Y$ be the first half of $X$ (i.e., $X=[Y;-\tilde{Y}]$) so that $\operatorname{n}(X)=[Y;-\tilde{Y};-Y;\tilde{Y}]$ is a palindrome. 8. 8.

If $X$ is symmetric and $Y$ is palindromic then $X\mathbin{\text{x}}Y$ is symmetric.

First, we show the even entries of $X\mathbin{\text{x}}Y$ satisfy the symmetric property. The $(2r)$th entry of $X\mathbin{\text{x}}Y$ is $x_{r}$ and the $(2n-2r)$th entry of $X\mathbin{\text{x}}Y$ is $x_{n-r}$ (for $r\neq 0$). Since $X$ is symmetric $x_{n-r}=x_{r}$ showing the $(2r)$th and $(2n-2r)$th entries are equal.

Second, we show the odd entries of $X\mathbin{\text{x}}Y$ satisfy the symmetric property. The $(2r+1)$th entry of $X\mathbin{\text{x}}Y$ is $y_{r}$ and the $(2n-2r-1)$the entry of $X\mathbin{\text{x}}Y$ is $y_{n-r-1}$. Since $Y$ is palindromic $y_{n-r-1}=y_{r}$ showing the $(2r+1)$th and $(2n-2r-1)$th entries are equal. 9. 9.

If $X$ is antisymmetric and of odd length then $\operatorname{n}(X)$ is symmetric.

Let $Y$ be the first half of the palindromic part of $X$ (i.e., $X=[x_{0};Y;-\tilde{Y}]$) so that $\operatorname{n}(X)=[x_{0};Y;-\tilde{Y};-x_{0};-Y;\tilde{Y}]$ is symmetric. 10. 10.

If $X$ is palindromic then $\operatorname{d}(X)$ is palindromic.

Note that $x_{2n-r-1\bmod n}=x_{n-r-1\bmod n}=x_{r\bmod n}$. Thus the $(2n-r-1)$th entry of $\operatorname{d}(X)$ is equal to the $r$th entry, as required.

VII-B List of odd perfect quaternion sequences

We now give palindromic odd perfect $Q_{+}$-sequences in all lengths $n<70$ except for $35$, $47$, $53$, $59$, $65$, and $67$. The sequences in odd lengths were constructed using Lemma 5 with a previously known set of Williamson sequences of length $n$. The sequences in even lengths were constructed using Lemma 6, Theorem 6, and Lemma 3 and are new to the best of our knowledge, though perfect quaternion sequences may be constructed in these lengths using the results of Barrera Acevedo and Dietrich [29].

The sequences are denoted by $P_{n}$ where $n$ is the length of the sequence. The symbols $\verb|+|$ and $\verb|-|$ denote $1$ and $-1$, capitalization denotes negation of an entry, and an overlined entry denotes left multiplication by $q$, i.e., the symbol \mathrlap{\overline{\color[rgb]{1,1,1}{\phantom{\mathtt{I}}}}}\verb|I| denotes the entry $-qi$.

$P_{1}=[\verb|+|]$

$P_{2}=[\verb|+|\verb|+|]$

P_{3}=[\verb|I|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|I|]

$P_{4}=[\verb|+|\verb|i|\verb|i|\verb|+|]$

P_{5}=[\verb|J|\verb|+|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|+|\verb|J|]

P_{6}=[\verb|i|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|I|\verb|I|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|i|]

P_{7}=[\verb|j|\verb|-|\verb|J|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|J|\verb|-|\verb|j|]

$P_{8}=[\verb|-|\verb|j|\verb|k|\verb|i|\verb|i|\verb|k|\verb|j|\verb|-|]$

P_{9}=[\verb|K|\verb|j|\verb|I|\verb|+|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|+|\verb|I|\verb|j|\verb|K|]

P_{10}=[\verb|J|\verb|-|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|+|\verb|J|\verb|J|\verb|+|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|-|\verb|J|]

P_{11}=[\verb|i|\verb|+|\verb|K|\verb|-|\verb|J|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|J|\verb|-|\verb|K|\verb|+|\verb|i|]

P_{12}=[\verb|+|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|i|\verb|+|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|K|\verb|I|\verb|I|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|K|\verb|+|\verb|i|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|+|]

P_{13}=[\verb|K|\verb|i|\verb|J|\verb|K|\verb|j|\verb|i|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|i|\verb|j|\verb|K|\verb|J|\verb|i|\verb|K|]

P_{14}=[\verb|J|\verb|-|\verb|J|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|j|\verb|-|\verb|j|\verb|j|\verb|-|\verb|j|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|J|\verb|-|\verb|J|]

P_{15}=[\verb|J|\verb|j|\verb|-|\verb|J|\verb|J|\verb|+|\verb|+|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|+|\verb|+|\verb|J|\verb|J|\verb|-|\verb|j|\verb|J|]

$P_{16}=[\verb|-|\verb|-|\verb|j|\verb|J|\verb|K|\verb|k|\verb|i|\verb|i|\verb|i|\verb|i|\verb|k|\verb|K|\verb|J|\verb|j|\verb|-|\verb|-|]$

P_{17}=[\verb|+|\verb|i|\verb|K|\verb|+|\verb|+|\verb|j|\verb|j|\verb|J|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|J|\verb|j|\verb|j|\verb|+|\verb|+|\verb|K|\verb|i|\verb|+|]

P_{18}=[\verb|K|\verb|J|\verb|i|\verb|+|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|-|\verb|i|\verb|j|\verb|K|\verb|K|\verb|j|\verb|i|\verb|-|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|+|\verb|i|\verb|J|\verb|K|]

P_{19}=[\verb|k|\verb|K|\verb|j|\verb|-|\verb|J|\verb|J|\verb|K|\verb|+|\verb|+|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|+|\verb|+|\verb|K|\verb|J|\verb|J|\verb|-|\verb|j|\verb|K|\verb|k|]

P_{20}=[\verb|k|\verb|-|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|i|\verb|K|\verb|j|\verb|+|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|k|\verb|i|\verb|J|\verb|J|\verb|i|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|k|\verb|+|\verb|j|\verb|K|\verb|i|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|-|\verb|k|]

P_{21}=[\verb|+|\verb|+|\verb|I|\verb|I|\verb|i|\verb|+|\verb|+|\verb|i|\verb|+|\verb|-|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|-|\verb|+|\verb|i|\verb|+|\verb|+|\verb|i|\verb|I|\verb|I|\verb|+|\verb|+|]

P_{22}=[\verb|I|\verb|+|\verb|K|\verb|+|\verb|j|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|J|\verb|+|\verb|k|\verb|+|\verb|i|\verb|i|\verb|+|\verb|k|\verb|+|\verb|J|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|j|\verb|+|\verb|K|\verb|+|\verb|I|]

P_{23}=[\verb|j|\verb|+|\verb|k|\verb|-|\verb|i|\verb|-|\verb|-|\verb|K|\verb|J|\verb|i|\verb|I|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|I|\verb|i|\verb|J|\verb|K|\verb|-|\verb|-|\verb|i|\verb|-|\verb|k|\verb|+|\verb|j|]

P_{24}=[\verb|+|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|J|\verb|k|\verb|I|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|k|\verb|j|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|k|\verb|-|\verb|J|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|I|\verb|i|\verb|i|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|I|\verb|J|\verb|-|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|k|\verb|j|\verb|k|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|I|\verb|k|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|J|\verb|+|]

P_{25}=[\verb|k|\verb|J|\verb|J|\verb|I|\verb|i|\verb|k|\verb|k|\verb|+|\verb|j|\verb|-|\verb|+|\verb|i|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|i|\verb|+|\verb|-|\verb|j|\verb|+|\verb|k|\verb|k|\verb|i|\verb|I|\verb|J|\verb|J|\verb|k|]

P_{26}=[\verb|K|\verb|I|\verb|j|\verb|K|\verb|j|\verb|I|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|i|\verb|j|\verb|k|\verb|j|\verb|i|\verb|K|\verb|K|\verb|i|\verb|j|\verb|k|\verb|j|\verb|i|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|I|\verb|j|\verb|K|\verb|j|\verb|I|\verb|K|]

P_{27}=[\verb|I|\verb|i|\verb|k|\verb|J|\verb|I|\verb|I|\verb|-|\verb|j|\verb|+|\verb|K|\verb|-|\verb|J|\verb|K|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|K|\verb|J|\verb|-|\verb|K|\verb|+|\verb|j|\verb|-|\verb|I|\verb|I|\verb|J|\verb|k|\verb|i|\verb|I|]

P_{28}=[\verb|k|\verb|-|\verb|j|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|k|\verb|k|\verb|+|\verb|j|\verb|K|\verb|i|\verb|j|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|K|\verb|I|\verb|j|\verb|j|\verb|I|\verb|K|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|j|\verb|i|\verb|K|\verb|j|\verb|+|\verb|k|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|k|\verb|j|\verb|-|\verb|k|]

P_{29}=[\verb|+|\verb|i|\verb|I|\verb|I|\verb|K|\verb|k|\verb|j|\verb|J|\verb|k|\verb|I|\verb|I|\verb|K|\verb|K|\verb|K|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|K|\verb|K|\verb|K|\verb|I|\verb|I|\verb|k|\verb|J|\verb|j|\verb|k|\verb|K|\verb|I|\verb|I|\verb|i|\verb|+|]

P_{30}=[\verb|j|\verb|j|\verb|-|\verb|j|\verb|j|\verb|+|\verb|+|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|-|\verb|+|\verb|J|\verb|j|\verb|+|\verb|j|\verb|J|\verb|J|\verb|j|\verb|+|\verb|j|\verb|J|\verb|+|\verb|-|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|+|\verb|+|\verb|j|\verb|j|\verb|-|\verb|j|\verb|j|]

P_{31}=[\verb|-|\verb|I|\verb|I|\verb|+|\verb|i|\verb|-|\verb|I|\verb|-|\verb|-|\verb|-|\verb|I|\verb|i|\verb|I|\verb|i|\verb|i|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|i|\verb|i|\verb|I|\verb|i|\verb|I|\verb|-|\verb|-|\verb|-|\verb|I|\verb|-|\verb|i|\verb|+|\verb|I|\verb|I|\verb|-|]

$P_{32}=[\verb|-|\verb|-|\verb|-|\verb|+|\verb|j|\verb|J|\verb|j|\verb|j|\verb|k|\verb|k|\verb|K|\verb|k|\verb|I|\verb|i|\verb|i|\verb|i|\verb|i|\verb|i|\verb|i|\verb|I|\verb|k|\verb|K|\verb|k|\verb|k|\verb|j|\verb|j|\verb|J|\verb|j|\verb|+|\verb|-|\verb|-|\verb|-|]$

P_{33}=[\verb|J|\verb|+|\verb|I|\verb|j|\verb|K|\verb|K|\verb|k|\verb|j|\verb|k|\verb|k|\verb|+|\verb|I|\verb|i|\verb|k|\verb|-|\verb|i|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|i|\verb|-|\verb|k|\verb|i|\verb|I|\verb|+|\verb|k|\verb|k|\verb|j|\verb|k|\verb|K|\verb|K|\verb|j|\verb|I|\verb|+|\verb|J|]

P_{34}=[\verb|+|\verb|I|\verb|k|\verb|+|\verb|+|\verb|J|\verb|J|\verb|J|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|j|\verb|J|\verb|j|\verb|+|\verb|-|\verb|k|\verb|i|\verb|+|\verb|+|\verb|i|\verb|k|\verb|-|\verb|+|\verb|j|\verb|J|\verb|j|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|J|\verb|J|\verb|J|\verb|+|\verb|+|\verb|k|\verb|I|\verb|+|]

P_{36}=[\verb|J|\verb|J|\verb|I|\verb|i|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|k|\verb|+|\verb|i|\verb|K|\verb|J|\verb|K|\verb|J|\verb|-|\verb|I|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|+|\verb|+|\verb|k|\verb|K|\verb|K|\verb|k|\verb|+|\verb|+|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|I|\verb|-|\verb|J|\verb|K|\verb|J|\verb|K|\verb|i|\verb|+|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|k|\verb|i|\verb|I|\verb|J|\verb|J|]

P_{37}=[\verb|j|\verb|i|\verb|k|\verb|I|\verb|I|\verb|J|\verb|I|\verb|+|\verb|K|\verb|k|\verb|J|\verb|-|\verb|J|\verb|J|\verb|+|\verb|-|\verb|K|\verb|+|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|+|\verb|K|\verb|-|\verb|+|\verb|J|\verb|J|\verb|-|\verb|J|\verb|k|\verb|K|\verb|+|\verb|I|\verb|J|\verb|I|\verb|I|\verb|k|\verb|i|\verb|j|]

P_{38}=[\verb|K|\verb|K|\verb|j|\verb|+|\verb|j|\verb|J|\verb|K|\verb|-|\verb|-|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|+|\verb|-|\verb|k|\verb|J|\verb|J|\verb|+|\verb|J|\verb|K|\verb|k|\verb|k|\verb|K|\verb|J|\verb|+|\verb|J|\verb|J|\verb|k|\verb|-|\verb|+|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|-|\verb|-|\verb|K|\verb|J|\verb|j|\verb|+|\verb|j|\verb|K|\verb|K|]

P_{39}=[\verb|J|\verb|k|\verb|k|\verb|-|\verb|J|\verb|+|\verb|I|\verb|+|\verb|K|\verb|k|\verb|J|\verb|+|\verb|+|\verb|+|\verb|k|\verb|j|\verb|K|\verb|J|\verb|j|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|j|\verb|J|\verb|K|\verb|j|\verb|k|\verb|+|\verb|+|\verb|+|\verb|J|\verb|k|\verb|K|\verb|+|\verb|I|\verb|+|\verb|J|\verb|-|\verb|k|\verb|k|\verb|J|]

P_{40}=[\verb|k|\verb|K|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|I|\verb|+|\verb|j|\verb|-|\verb|K|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|K|\verb|I|\verb|I|\verb|+|\verb|-|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|J|\verb|I|\verb|K|\verb|i|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|j|\verb|j|\verb|j|\verb|j|\verb|j|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|j|\verb|i|\verb|K|\verb|I|\verb|J|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|-|\verb|+|\verb|I|\verb|I|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|K|\verb|K|\verb|-|\verb|j|\verb|+|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|I|\verb|K|\verb|k|]

P_{41}=[\verb|i|\verb|K|\verb|k|\verb|K|\verb|i|\verb|k|\verb|I|\verb|I|\verb|K|\verb|K|\verb|i|\verb|k|\verb|i|\verb|K|\verb|i|\verb|i|\verb|I|\verb|i|\verb|k|\verb|k|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|k|\verb|k|\verb|i|\verb|I|\verb|i|\verb|i|\verb|K|\verb|i|\verb|k|\verb|i|\verb|K|\verb|K|\verb|I|\verb|I|\verb|k|\verb|i|\verb|K|\verb|k|\verb|K|\\ \phantom{P_{00}=[}\verb|i|]

P_{42}=[\verb|+|\verb|-|\verb|i|\verb|I|\verb|i|\verb|-|\verb|-|\verb|i|\verb|+|\verb|+|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|-|\verb|+|\verb|I|\verb|-|\verb|+|\verb|i|\verb|i|\verb|i|\verb|+|\verb|+|\verb|+|\verb|+|\verb|i|\verb|i|\verb|i|\verb|+|\verb|-|\verb|I|\verb|+|\verb|-|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|+|\verb|+|\verb|i|\verb|-|\verb|-|\verb|i|\verb|I|\verb|i|\\ \phantom{P_{00}=[}\verb|-|\verb|+|]

P_{43}=[\verb|j|\verb|-|\verb|I|\verb|J|\verb|J|\verb|j|\verb|I|\verb|K|\verb|K|\verb|K|\verb|+|\verb|j|\verb|+|\verb|k|\verb|-|\verb|+|\verb|K|\verb|i|\verb|j|\verb|K|\verb|-|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|-|\verb|K|\verb|j|\verb|i|\verb|K|\verb|+|\verb|-|\verb|k|\verb|+|\verb|j|\verb|+|\verb|K|\verb|K|\verb|K|\verb|I|\verb|j|\verb|J|\verb|J|\\ \phantom{P_{00}=[}\verb|I|\verb|-|\verb|j|]

P_{44}=[\verb|-|\verb|+|\verb|k|\verb|i|\verb|k|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|J|\verb|I|\verb|j|\verb|+|\verb|I|\verb|-|\verb|i|\verb|K|\verb|+|\verb|k|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|k|\verb|j|\verb|-|\verb|j|\verb|i|\verb|i|\verb|i|\verb|i|\verb|j|\verb|-|\verb|j|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|k|\verb|k|\verb|+|\verb|K|\verb|i|\verb|-|\verb|I|\verb|+|\verb|j|\verb|I|\verb|J|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|k|\\ \phantom{P_{00}=[}\verb|i|\verb|k|\verb|+|\verb|-|]

P_{45}=[\verb|k|\verb|k|\verb|k|\verb|K|\verb|k|\verb|I|\verb|i|\verb|K|\verb|I|\verb|k|\verb|i|\verb|k|\verb|K|\verb|K|\verb|I|\verb|K|\verb|I|\verb|i|\verb|K|\verb|K|\verb|i|\verb|i|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|i|\verb|i|\verb|K|\verb|K|\verb|i|\verb|I|\verb|K|\verb|I|\verb|K|\verb|K|\verb|k|\verb|i|\verb|k|\verb|I|\verb|K|\verb|i|\verb|I|\\ \phantom{P_{00}=[}\verb|k|\verb|K|\verb|k|\verb|k|\verb|k|]

P_{46}=[\verb|J|\verb|+|\verb|k|\verb|+|\verb|I|\verb|-|\verb|-|\verb|k|\verb|j|\verb|i|\verb|I|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|i|\verb|i|\verb|J|\verb|k|\verb|+|\verb|-|\verb|i|\verb|+|\verb|K|\verb|+|\verb|j|\verb|j|\verb|+|\verb|K|\verb|+|\verb|i|\verb|-|\verb|+|\verb|k|\verb|J|\verb|i|\verb|i|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|I|\verb|i|\verb|j|\verb|k|\verb|-|\\ \phantom{P_{00}=[}\verb|-|\verb|I|\verb|+|\verb|k|\verb|+|\verb|J|]

P_{48}=[\verb|+|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|K|\verb|J|\verb|J|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|I|\verb|K|\verb|i|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|I|\verb|i|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|I|\verb|K|\verb|J|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|j|\verb|-|\verb|+|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|K|\verb|+|\verb|j|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|J|\verb|k|\verb|k|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|i|\verb|i|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|k|\verb|k|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|J|\verb|j|\verb|+|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|K|\verb|+|\verb|-|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|j|\verb|J|\verb|K|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|I|\verb|i|\verb|I|\\ \phantom{P_{00}=[}\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|i|\verb|K|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|I|\verb|J|\verb|J|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|K|\verb|+|]

P_{49}=[\verb|k|\verb|k|\verb|i|\verb|I|\verb|I|\verb|k|\verb|I|\verb|I|\verb|i|\verb|I|\verb|I|\verb|I|\verb|k|\verb|i|\verb|K|\verb|k|\verb|i|\verb|k|\verb|k|\verb|I|\verb|I|\verb|K|\verb|k|\verb|K|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|K|\verb|k|\verb|K|\verb|I|\verb|I|\verb|k|\verb|k|\verb|i|\verb|k|\verb|K|\verb|i|\verb|k|\verb|I|\verb|I|\verb|I|\\ \phantom{P_{00}=[}\verb|i|\verb|I|\verb|I|\verb|k|\verb|I|\verb|I|\verb|i|\verb|k|\verb|k|]

P_{50}=[\verb|k|\verb|j|\verb|j|\verb|I|\verb|i|\verb|K|\verb|K|\verb|+|\verb|j|\verb|+|\verb|-|\verb|i|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|I|\verb|-|\verb|-|\verb|j|\verb|-|\verb|K|\verb|k|\verb|i|\verb|i|\verb|j|\verb|J|\verb|k|\verb|k|\verb|J|\verb|j|\verb|i|\verb|i|\verb|k|\verb|K|\verb|-|\verb|j|\verb|-|\verb|-|\verb|I|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|i|\verb|-|\\ \phantom{P_{00}=[}\verb|+|\verb|j|\verb|+|\verb|K|\verb|K|\verb|i|\verb|I|\verb|j|\verb|j|\verb|k|]

P_{51}=[\verb|J|\verb|J|\verb|-|\verb|j|\verb|+|\verb|-|\verb|+|\verb|-|\verb|j|\verb|+|\verb|+|\verb|j|\verb|+|\verb|J|\verb|+|\verb|+|\verb|+|\verb|j|\verb|-|\verb|j|\verb|J|\verb|-|\verb|-|\verb|+|\verb|j|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|j|\verb|+|\verb|-|\verb|-|\verb|J|\verb|j|\verb|-|\verb|j|\verb|+|\verb|+|\verb|+|\verb|J|\verb|+|\verb|j|\\ \phantom{P_{00}=[}\verb|+|\verb|+|\verb|j|\verb|-|\verb|+|\verb|-|\verb|+|\verb|j|\verb|-|\verb|J|\verb|J|]

P_{52}=[\verb|J|\verb|I|\verb|J|\verb|j|\verb|k|\verb|i|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|+|\verb|K|\verb|k|\verb|J|\verb|-|\verb|j|\verb|k|\verb|i|\verb|K|\verb|j|\verb|j|\verb|I|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|K|\verb|+|\verb|J|\verb|K|\verb|K|\verb|-|\verb|K|\verb|K|\verb|-|\verb|K|\verb|K|\verb|J|\verb|+|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|K|\verb|I|\verb|j|\verb|j|\verb|K|\verb|i|\verb|k|\verb|j|\\ \phantom{P_{00}=[}\verb|-|\verb|J|\verb|k|\verb|K|\verb|+|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|i|\verb|k|\verb|j|\verb|J|\verb|I|\verb|J|]

P_{54}=[\verb|i|\verb|i|\verb|k|\verb|j|\verb|i|\verb|I|\verb|-|\verb|J|\verb|-|\verb|K|\verb|-|\verb|j|\verb|k|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|K|\verb|j|\verb|+|\verb|K|\verb|+|\verb|J|\verb|+|\verb|I|\verb|I|\verb|j|\verb|K|\verb|i|\verb|I|\verb|I|\verb|i|\verb|K|\verb|j|\verb|I|\verb|I|\verb|+|\verb|J|\verb|+|\verb|K|\verb|+|\verb|j|\verb|K|\\ \phantom{P_{00}=[}\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|k|\verb|j|\verb|-|\verb|K|\verb|-|\verb|J|\verb|-|\verb|I|\verb|i|\verb|j|\verb|k|\verb|i|\verb|i|]

P_{55}=[\verb|J|\verb|J|\verb|J|\verb|+|\verb|+|\verb|j|\verb|J|\verb|+|\verb|+|\verb|-|\verb|+|\verb|-|\verb|+|\verb|+|\verb|j|\verb|j|\verb|-|\verb|j|\verb|+|\verb|-|\verb|J|\verb|+|\verb|-|\verb|-|\verb|j|\verb|+|\verb|j|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|j|\verb|+|\verb|j|\verb|-|\verb|-|\verb|+|\verb|J|\verb|-|\verb|+|\verb|j|\verb|-|\verb|j|\\ \phantom{P_{00}=[}\verb|j|\verb|+|\verb|+|\verb|-|\verb|+|\verb|-|\verb|+|\verb|+|\verb|J|\verb|j|\verb|+|\verb|+|\verb|J|\verb|J|\verb|J|]

P_{56}=[\verb|k|\verb|K|\verb|+|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|J|\verb|j|\verb|i|\verb|k|\verb|I|\verb|i|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|I|\verb|j|\verb|-|\verb|+|\verb|I|\verb|I|\verb|K|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|J|\verb|+|\verb|+|\verb|j|\verb|-|\verb|K|\verb|K|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|K|\verb|i|\verb|J|\verb|J|\verb|J|\verb|J|\verb|i|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|K|\verb|K|\verb|K|\verb|-|\verb|j|\verb|+|\verb|+|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|J|\verb|K|\\ \phantom{P_{00}=[}\verb|I|\verb|I|\verb|+|\verb|-|\verb|j|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|I|\verb|i|\verb|I|\verb|k|\verb|i|\verb|j|\verb|J|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|+|\verb|K|\verb|k|]

P_{57}=[\verb|I|\verb|i|\verb|+|\verb|+|\verb|I|\verb|+|\verb|-|\verb|+|\verb|+|\verb|-|\verb|i|\verb|i|\verb|I|\verb|-|\verb|-|\verb|-|\verb|+|\verb|i|\verb|+|\verb|I|\verb|-|\verb|i|\verb|-|\verb|-|\verb|I|\verb|I|\verb|-|\verb|I|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|I|\verb|-|\verb|I|\verb|I|\verb|-|\verb|-|\verb|i|\verb|-|\verb|I|\verb|+|\verb|i|\\ \phantom{P_{00}=[}\verb|+|\verb|-|\verb|-|\verb|-|\verb|I|\verb|i|\verb|i|\verb|-|\verb|+|\verb|+|\verb|-|\verb|+|\verb|I|\verb|+|\verb|+|\verb|i|\verb|I|]

P_{58}=[\verb|+|\verb|I|\verb|i|\verb|I|\verb|K|\verb|K|\verb|J|\verb|J|\verb|k|\verb|i|\verb|i|\verb|K|\verb|K|\verb|k|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|K|\verb|K|\verb|k|\verb|i|\verb|I|\verb|k|\verb|j|\verb|J|\verb|k|\verb|K|\verb|i|\verb|i|\verb|i|\verb|+|\verb|+|\verb|i|\verb|i|\verb|i|\verb|K|\verb|k|\verb|J|\verb|j|\verb|k|\verb|I|\verb|i|\\ \phantom{P_{00}=[}\verb|k|\verb|K|\verb|K|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|k|\verb|K|\verb|K|\verb|i|\verb|i|\verb|k|\verb|J|\verb|J|\verb|K|\verb|K|\verb|I|\verb|i|\verb|I|\verb|+|]

P_{60}=[\verb|K|\verb|j|\verb|+|\verb|k|\verb|k|\verb|-|\verb|+|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|K|\verb|i|\verb|+|\verb|j|\verb|k|\verb|i|\verb|J|\verb|J|\verb|k|\verb|K|\verb|+|\verb|J|\verb|K|\verb|i|\verb|+|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|I|\verb|I|\verb|j|\verb|J|\verb|I|\verb|k|\verb|J|\verb|J|\verb|k|\verb|I|\verb|J|\verb|j|\verb|I|\verb|I|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|+|\verb|i|\\ \phantom{P_{00}=[}\verb|K|\verb|J|\verb|+|\verb|K|\verb|k|\verb|J|\verb|J|\verb|i|\verb|k|\verb|j|\verb|+|\verb|i|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|K|\verb|+|\verb|-|\verb|k|\verb|k|\verb|+|\verb|j|\verb|K|]

P_{61}=[\verb|K|\verb|k|\verb|K|\verb|K|\verb|I|\verb|i|\verb|K|\verb|i|\verb|K|\verb|I|\verb|I|\verb|k|\verb|k|\verb|K|\verb|i|\verb|I|\verb|I|\verb|K|\verb|I|\verb|I|\verb|K|\verb|k|\verb|i|\verb|K|\verb|K|\verb|i|\verb|i|\verb|i|\verb|I|\verb|K|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|K|\verb|I|\verb|i|\verb|i|\verb|i|\verb|K|\verb|K|\verb|i|\verb|k|\\ \phantom{P_{00}=[}\verb|K|\verb|I|\verb|I|\verb|K|\verb|I|\verb|I|\verb|i|\verb|K|\verb|k|\verb|k|\verb|I|\verb|I|\verb|K|\verb|i|\verb|K|\verb|i|\verb|I|\verb|K|\verb|K|\verb|k|\verb|K|]

P_{62}=[\verb|+|\verb|I|\verb|I|\verb|-|\verb|I|\verb|-|\verb|I|\verb|+|\verb|+|\verb|-|\verb|I|\verb|I|\verb|i|\verb|i|\verb|i|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|I|\verb|i|\verb|I|\verb|I|\verb|i|\verb|-|\verb|-|\verb|+|\verb|i|\verb|-|\verb|i|\verb|-|\verb|i|\verb|I|\verb|-|\verb|-|\verb|I|\verb|i|\verb|-|\verb|i|\verb|-|\verb|i|\verb|+|\verb|-|\\ \phantom{P_{00}=[}\verb|-|\verb|i|\verb|I|\verb|I|\verb|i|\verb|I|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|i|\verb|i|\verb|i|\verb|I|\verb|I|\verb|-|\verb|+|\verb|+|\verb|I|\verb|-|\verb|I|\verb|-|\verb|I|\verb|I|\verb|+|]

P_{63}=[\verb|-|\verb|i|\verb|i|\verb|i|\verb|I|\verb|i|\verb|+|\verb|j|\verb|j|\verb|j|\verb|J|\verb|+|\verb|k|\verb|i|\verb|K|\verb|J|\verb|-|\verb|K|\verb|k|\verb|-|\verb|i|\verb|k|\verb|-|\verb|I|\verb|j|\verb|-|\verb|-|\verb|-|\verb|+|\verb|i|\verb|I|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|-|\verb|I|\verb|i|\verb|+|\verb|-|\verb|-|\verb|-|\verb|j|\verb|I|\\ \phantom{P_{00}=[}\verb|-|\verb|k|\verb|i|\verb|-|\verb|k|\verb|K|\verb|-|\verb|J|\verb|K|\verb|i|\verb|k|\verb|+|\verb|J|\verb|j|\verb|j|\verb|j|\verb|+|\verb|i|\verb|I|\verb|i|\verb|i|\verb|i|\verb|-|]

$P_{64}=[\verb|-|\verb|-|\verb|-|\verb|+|\verb|-|\verb|-|\verb|+|\verb|-|\verb|j|\verb|J|\verb|j|\verb|j|\verb|j|\verb|J|\verb|J|\verb|J|\verb|K|\verb|K|\verb|K|\verb|k|\verb|k|\verb|k|\verb|K|\verb|k|\verb|i|\verb|I|\verb|i|\verb|i|\verb|I|\verb|i|\verb|i|\verb|i|\verb|i|\verb|i|\verb|i|\verb|I|\verb|i|\verb|i|\verb|I|\verb|i|\\ \phantom{P_{00}=[}\verb|k|\verb|K|\verb|k|\verb|k|\verb|k|\verb|K|\verb|K|\verb|K|\verb|J|\verb|J|\verb|J|\verb|j|\verb|j|\verb|j|\verb|J|\verb|j|\verb|-|\verb|+|\verb|-|\verb|-|\verb|+|\verb|-|\verb|-|\verb|-|]$

P_{66}=[\verb|J|\verb|-|\verb|i|\verb|j|\verb|K|\verb|k|\verb|K|\verb|j|\verb|k|\verb|K|\verb|-|\verb|I|\verb|i|\verb|K|\verb|+|\verb|i|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|I|\verb|+|\verb|k|\verb|i|\verb|i|\verb|-|\verb|k|\verb|k|\verb|J|\verb|K|\verb|K|\verb|K|\verb|J|\verb|i|\verb|+|\verb|J|\verb|J|\verb|+|\verb|i|\verb|J|\verb|K|\verb|K|\verb|K|\\ \phantom{P_{00}=[}\verb|J|\verb|k|\verb|k|\verb|-|\verb|i|\verb|i|\verb|k|\verb|+|\verb|I|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|i|\verb|+|\verb|K|\verb|i|\verb|I|\verb|-|\verb|K|\verb|k|\verb|j|\verb|K|\verb|k|\verb|K|\verb|j|\verb|i|\verb|-|\verb|J|]

P_{68}=[\verb|I|\verb|I|\verb|K|\verb|i|\verb|i|\verb|j|\verb|J|\verb|k|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|K|\verb|j|\verb|j|\verb|k|\verb|i|\verb|+|\verb|k|\verb|+|\verb|I|\verb|+|\verb|I|\verb|J|\verb|I|\verb|-|\verb|j|\verb|k|\verb|K|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|j|\verb|K|\verb|K|\verb|-|\verb|+|\verb|j|\verb|-|\verb|+|\verb|+|\verb|-|\verb|j|\verb|+|\verb|-|\verb|K|\\ \phantom{P_{00}=[}\verb|K|\verb|j|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|K|\verb|k|\verb|j|\verb|-|\verb|I|\verb|J|\verb|I|\verb|+|\verb|I|\verb|+|\verb|k|\verb|+|\verb|i|\verb|k|\verb|j|\verb|j|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|K|\verb|k|\verb|J|\verb|j|\verb|i|\verb|i|\verb|K|\verb|I|\verb|I|]

P_{69}=[\verb|-|\verb|-|\verb|-|\verb|-|\verb|-|\verb|j|\verb|-|\verb|J|\verb|J|\verb|j|\verb|-|\verb|-|\verb|+|\verb|j|\verb|-|\verb|+|\verb|-|\verb|J|\verb|J|\verb|J|\verb|j|\verb|-|\verb|j|\verb|+|\verb|J|\verb|j|\verb|-|\verb|j|\verb|j|\verb|-|\verb|J|\verb|j|\verb|+|\verb|-|\mathrlap{\overline{\color[rgb]{1,1,1}{\vphantom{I}\mathtt{q}}}}\verb|+|\verb|-|\verb|+|\verb|j|\verb|J|\verb|-|\\ \phantom{P_{00}=[}\verb|j|\verb|j|\verb|-|\verb|j|\verb|J|\verb|+|\verb|j|\verb|-|\verb|j|\verb|J|\verb|J|\verb|J|\verb|-|\verb|+|\verb|-|\verb|j|\verb|+|\verb|-|\verb|-|\verb|j|\verb|J|\verb|J|\verb|-|\verb|j|\verb|-|\verb|-|\verb|-|\verb|-|\verb|-|]

Acknowledgment

The authors would like to thank the editors and reviewers for their careful proofreading of our work.