Circulant q-Butson Hadamard matrices

TL;DR

This paper investigates the existence and construction of circulant q-Butson Hadamard matrices, providing new constraints on their dimensions and explicit examples, advancing understanding in combinatorial matrix theory.

Contribution

It establishes a strong algebraic number theory-based constraint on the dimension of circulant q-Butson Hadamard matrices when dimensions are prime powers and constructs explicit examples in all such dimensions.

Findings

Derived a dimension constraint for circulant q-Butson Hadamard matrices

Constructed explicit examples in all prime power dimensions

Connected results to the circulant Hadamard matrix conjecture

Abstract

If $q = p^n$ is a prime power, then a $d$-dimensional \emph{$q$-Butson Hadamard matrix} $H$ is a $d\times d$ matrix with all entries $q$th roots of unity such that $HH^* = dI_d$. We use algebraic number theory to prove a strong constraint on the dimension of a circulant $q$-Butson Hadamard matrix when $d = p^m$ and then explicitly construct a family of examples in all possible dimensions. These results relate to the long-standing circulant Hadamard matrix conjecture in combinatorics.

Equations

$$H{H}^{\ast }=d{I}_d,$$

$$H_{ij}=f(i-j)$$

$$\big{(}H_{ij}\big{)}=\big{(}\zeta^{f(i-j)}\big{)}.$$

$$H{H}^{\ast }=d{I}_d,$$

$$C_{ij}=f(i-j)$$

$$\left(\begin{array}{ccc}[l]1& 1& 1\\ 1& \omega & {\omega }^2\\ 1& {\omega }^2& {\omega }^4\end{array}\right)$$

$$\left(\begin{array}{ccc}1& \omega & \omega \\ \omega & 1& \omega \\ \omega & \omega & 1\end{array}\right)$$

$$H{H}^{\ast }=d{I}_d,$$

$${\alpha }_k=\sum _{j<d}{a}_j{\zeta }^{jk}.$$

$$u_{k}^{t}=\big{(}1,\zeta^{k},\zeta^{2k},\ldots,\zeta^{(d-1)k}\big{)}.$$

$$\prod _k{\alpha }_k=\det (H)=\pm p^{(m+n)d/2}.$$

$${\alpha }_0=\pm {\zeta }^r{\alpha }_1.$$

$${\alpha }_0=\sum _{j<d}{a}_j{\alpha }_1=\sum _{j<d}{a}_j{\zeta }^j.$$

$${\zeta }^j={\zeta }^{j_0+j_1{p}^m}={\omega }^{j_1}{\zeta }^{j_0}.$$

$${\alpha }_1=\sum _{j<d}{a}_j{\zeta }^j=\sum _{j^{\prime }<p^m}{b}_{j^{\prime }}{\zeta }^{j^{\prime }},$$

$$\sum _{j<d}{a}_j={\alpha }_0=\pm {\omega }^{j_0}{\zeta }^{-j_1}{\alpha }_1=\sum _{j^{\prime }<p^m}\pm {\omega }^{j_0}{b}_{j^{\prime }}{\zeta }^{j^{\prime }-j_1}.$$

$${\alpha }_0=\pm {\omega }^{j_0}{b}_{j_1},$$

$$\sum _{x\in X}{\zeta }^{g(x)}=0.$$

$${\Phi }_{p^n}(x)=\sum _{j<p}{x}^{j{p}^{n-1}}$$

$$s(x)=\sum _{i<p^{n-1}}{a}_i{x}^i$$

$$r(x)=s(x)\Phi_{p^{n}}(x)=\sum_{\vbox{\Let@\restore@math@cr\default@tag\halign{\hfil$\m@th\scriptstyle#$&$\m@th\scriptstyle{}#$\cr i&<p^{n-1}\\ j&< p\crcr}}}a_{i}x^{i+jp^{n-1}}.$$

$$b_k=b_{i+j{p}^{n-1}}=a_i,$$

$$\sum _{x\in X}{\zeta }^{g(x)}=\sum _{i<p^{n-1}}\sum _{j<p}{a}_i{\zeta }^{i+j{p}^{n-1}}={\Phi }_{p^n}(\zeta )\sum _{i<p^{n-1}}{a}_i{\zeta }^i=0.$$

$$0=\sum _{x\in X}{\zeta }^{g(x)}=\sum _{k<p^n}{c}_k{\zeta }^k,$$

$$\big{(}H_{i,j}\big{)}=\big{(}\zeta^{f(i-j)}\big{)}.$$

$$r_{j,j+k}=\sum _{i<d}{\zeta }^{f(i-j+k)-f(i-j)}=\sum _{i<d}{\zeta }^{{\delta }_k(i-j)}=0,$$

$$0=r_{j,j+k}=\sum _{i<d}{\zeta }^{f(i-j+k)-f(i-j)}.$$

$$\sum _{j<q}{\zeta }^{f(j)}=\sum _{j<q}{\zeta }^{aj+b}={\zeta }^b\sum _{j<q}({\zeta }^a{)}^j=0.$$

$${\delta }_k:\mathbb{Z}/(p^m)\times \mathbb{Z}/(q)\to \mathbb{Z}/(q)$$

$${\delta }_k(x,y)=ky+\sum _{j<k}{S}_j(x),$$

Full text

1. Introduction

For a prime power $q=p^{n}$, a $q$-Butson Hadamard matrix ($q$-BH) of dimension $d$ is a $d\times d$ matrix $H$ with all entries $q$th roots of unity such that

[TABLE]

where $H^{*}$ is the conjugate transpose of $H$. A $d\times d$ matrix $H$ is said to be circulant if

[TABLE]

for some function $f$ defined modulo $d$. In this paper we investigate circulant $q$-BH matrices of dimension $d=p^{m}$.

Theorem 1**.**

If $q=p^{n}$ is a prime power and $d=p^{m}$, then there exists a $d\times d$ circulant $q$-Butson Hadamard matrix if and only if $m\leq 2n$, with one exception when $(m,n,p)=(1,1,2)$.

Our analysis of circulant $q$-BH matrices led us to the useful notion of fibrous functions.

Definition 2**.**

Let $d\geq 0$ and $q=p^{n}$ be a prime power,

1. (1)

If $X$ is a finite set, we say a function $g:X\rightarrow\mathbb{Z}/(q)$ is fibrous if the cardinality of the fiber $|g^{-1}(b)|$ depends only on $b\bmod p^{n-1}$. 2. (2)

We say a function $f:\mathbb{Z}/(d)\rightarrow\mathbb{Z}/(q)$ is $\delta$-fibrous if for each $k\not\equiv 0\bmod d$ the function $\delta_{k}(x)=f(x+k)-f(x)$ is fibrous.

When $q=d=p$ are both prime, $\delta$-fibrous functions coincide with the concept of planar functions, which arise in the study of finite projective planes [2] and have applications in cryptography [6]. Circulant $q$-BH matrices of dimension $d$ are equivalent to $\delta$-fibrous functions $f:\mathbb{Z}/(d)\rightarrow\mathbb{Z}/(q)$.

Theorem 3**.**

Let $q=p^{n}$ be a prime power and $\zeta$ a primitive $q$th root of unity. There is a correspondence between $d\times d$ circulant $q$-Butson Hadamard matrices $H$ and $\delta$-fibrous functions $f:\mathbb{Z}/(d)\to\mathbb{Z}/(q)$ given by

[TABLE]

We restate our main result in the language of $\delta$-fibrous functions.

Corollary 4**.**

If $q=p^{n}$ is a prime power, then there exist $\delta$-fibrous functions $f:\mathbb{Z}/(p^{m})\rightarrow\mathbb{Z}/(q)$ if and only if $m\leq 2n$ with one exception when $(m,n,p)=(1,1,2)$.

It would be interesting to know if $\delta$-fibrous functions have applications to finite geometry or cryptography.

When $q=2$, $q$-BH matrices are called Hadamard matrices. Hadamard matrices are usually defined as $d\times d$ matrices $H$ with all entries $\pm 1$ such that $HH^{t}=dI_{d}$. Buston Hadamard matrices were introduced in [1] as a generalization of Hadamard matrices.

Circulant Hadamard matrices arise in the theory of difference sets, combinatorial designs, and synthetic geometry [7, Chap. 9]. There are arithmetic constraints on the possible dimension of a circulant Hadamard matrix. When the dimension $d=2^{m}$ is a power of two we have:

Theorem 5** (Turyn [9]).**

If $d=2^{m}$ is the dimension of a circulant Hadamard matrix, then $m=0$ or $m=2$.

Turyn’s proof uses algebraic number theory, more specifically the fact that 2 is totally ramified in the $2^{n}$th cyclotomic extension $\mathbb{Q}(\zeta_{2^{n}})/\mathbb{Q}$; an elementary exposition may be found in Stanley [8]. Conjecturally this accounts for all circulant Hadamard matrices [7].

Conjecture 6**.**

There are no $d$-dimensional circulant Hadamard matrices for $d>4$.

Circulant $q$-Butson Hadamard matrices provide a natural context within which to consider circulant Hadamard matrices. A better understanding of the former could lead to new insights on the latter. For example, our proof of Theorem 1 shows that the two possible dimensions for a circulant Hadamard matrix given by Turyn’s theorem belong to larger family of circulant $q$-BH matrices with the omission of $d=2$ being a degenerate exception. Circulant $q$-BH matrices have been studied in some specific dimensions [3], but overall seem poorly understood. We leave the existence of $q$-BH matrices when the dimension $d$ is not a power of $p$ for future work.

Acknowledgements

The authors thank Padraig Cathain for pointers to the literature, in particular for bringing the work of de Launey [4] to our attention. We also thank Jeff Lagarias for helpful feedback on an earlier draft.

2. Main Results

We always let $q=p^{n}$ denote a prime power. First recall the definitions of $q$-Butson Hadamard and circulant matrices.

Definition 7**.**

A $d$-dimensional $q$-Butson Hadamard matrix $H$ ($q$-BH) is a $d\times d$ matrix all of whose entries are $q$th roots of unity satisfying

[TABLE]

*where $H^{*}$ is the conjugate transpose of $H$.

A $d$-dimensional circulant matrix $C$ is a $d\times d$ matrix with coefficients in a ring $R$ such that

[TABLE]

for some function $f:\mathbb{Z}/(d)\rightarrow R$.

Example 8**.**

Hadamard matrices are the special case of $q$-BH matrices with $q=2$. The $q$-Fourier matrix $(\zeta^{ij})$ where $\zeta$ is a primitive $q$th root of unity is an example of a $q$-BH matrix (which may also be interpreted as the character table of the cyclic group $\mathbb{Z}/(q)$.) When $q=3$ and $\omega$ is a primitive 3rd root of unity this is the matrix:

[TABLE]

This example is not a circulant $3$-BH matrix. The following is a circulant $3$-BH matrix:

[TABLE]

The remainder of the paper is divided into two sections: first we prove constraints on the dimension $d$ of a circulant $q$-BH matrix when $d$ is a power of $p$; next we introduce the concept of $\delta$-fibrous functions and construct examples of circulant $q$-BH matrices in all possible dimensions.

Constraints on dimension

Theorem 9 uses the ramification of the prime $p$ in the $q$th cyclotomic extension $\mathbb{Q}(\zeta)/\mathbb{Q}$ to deduce strong constraints on the dimension of a $q$-BH matrix.

Theorem 9**.**

If $q=p^{n}$ is a prime power and $H$ is a circulant $q$-Butson Hadamard matrix of dimension $d=p^{m+n}$, then $m\leq n$.

Note that our indexing of $m$ has changed from the introduction; this choice was made to improve notation in our proof. We use this indexing for the rest of the paper.

Proof of Theorem 9.

Suppose $H=(a_{i-j})$ is a circulant $q$-Butson Hadamard matrix of dimension $d=p^{m+n}$. From $H$ being $q$-BH of dimension $d$ we have

[TABLE]

hence $\det(H)=\pm d^{d/2}$ and each eigenvalue $\alpha$ of $H$ has absolute value $|\alpha|=\sqrt{d}=p^{(m+n)/2}$. On the other hand, since $H=(a_{i-j})$ is circulant, it has eigenvalues

[TABLE]

for $\zeta$ a primitive $d$th root of unity with corresponding eigenvector

[TABLE]

These observations combine to give two ways of computing $\det(H)$.

[TABLE]

The identity (2) is the essential interaction between the circulant and $q$-BH conditions on $H$. The prime $p$ is totally ramified in $\mathbb{Q}(\zeta)$, hence there is a unique prime ideal $\mathfrak{p}\subseteq\mathbb{Z}[\zeta]$ over $(p)\subseteq\mathbb{Z}$. Since all $\alpha_{k}\in\mathbb{Z}[\zeta]$, it follows from (2) that $(\alpha_{k})=\mathfrak{p}^{v_{k}}$ as ideals of $\mathbb{Z}[\zeta]$ for some $v_{k}\geq 0$ and for each $k$. So either $\alpha_{0}/\alpha_{1}$ or $\alpha_{1}/\alpha_{0}$ is an element of $\mathbb{Z}[\zeta]$. Say $\alpha_{0}/\alpha_{1}$ is the integral quotient. We noted $|\alpha_{k}|=p^{(m+n)/2}$ for each $k$, hence $|\alpha_{0}/\alpha_{1}|=1$. The only integral elements of $\mathbb{Z}[\zeta]$ with absolute value 1 are roots of unity, hence $\alpha_{0}/\alpha_{1}=\pm\zeta^{r}$ for some $r\geq 0$, hence

[TABLE]

By (1) we have

[TABLE]

Each $j<d=p^{m+n}$ has a unique expression as $j=j_{0}+j_{1}p^{m}$ where $j_{0}<p^{m}$ and $j_{1}<p^{n}$. Let $\omega=\zeta^{p^{m}}$ be a primitive $q$th root of unity. Then

[TABLE]

Writing $\alpha_{1}$ in the linear basis \big{\{}1,\zeta,\zeta^{2},\ldots,\zeta^{p^{m}-1}\big{\}} of $\mathbb{Q}(\zeta)/\mathbb{Q}(\omega)$ we have

[TABLE]

where $b_{j^{\prime}}$ is a sum of $p^{n}$ complex numbers each with absolute value 1. Now (3) says $\alpha_{0}=\pm\omega^{j_{0}}\zeta^{-j_{1}}\alpha_{1}$ for some $j_{0}$ and $j_{1}$, thus

[TABLE]

Comparing coefficients we conclude that

[TABLE]

which is to say that $\alpha_{0}$ is the sum of $p^{n}$ complex numbers each with absolute value 1, hence $|\alpha_{0}|\leq p^{n}$. On the other hand we have $|\alpha_{0}|=p^{(m+n)/2}$. Thus $m+n\leq 2n\Longrightarrow 0\leq m\leq n$ as desired. ∎

Remark*.*

The main impediment to extending this result from $q=p^{n}$ to a general integer $q$ is that we no longer have the total ramification of the primes dividing the determinant of $H$. It may be possible to get some constraint in certain cases from a closer analysis of the eigenvalues $\alpha_{k}$ and ramification, but we do not pursue this.

$\delta$-Fibrous functions and construction of circulant $q$-BH matrices

Recall the notion of fibrous functions from the introduction:

Definition 10**.**

Let $d\geq 0$ and $q=p^{n}$ be a prime power,

1. (1)

If $X$ is a finite set, we say $g:X\rightarrow\mathbb{Z}/(q)$ is fibrous if the cardinality of the fibers $|g^{-1}(b)|$ depends only on $b\bmod p^{n-1}$. 2. (2)

We say a function $f:\mathbb{Z}/(d)\rightarrow\mathbb{Z}/(q)$ is $\delta$-fibrous if for each $k\not\equiv 0\bmod d$ the function $x\mapsto f(x+k)-f(x)$ is fibrous.

Lemma 11 is a combinatorial reinterpretation of the cyclotomic polynomials $\Phi_{p^{n}}(x)$.

Lemma 11**.**

Let $p$ be a prime and $\zeta$ a primitive $p^{n}$th root of unity.

1. (1)

If $\sum_{k<p^{n}}b_{k}\zeta^{k}=0$ with $b_{k}\in\mathbb{Q}$, then $b_{k}$ depends only on $k\bmod p^{n-1}$. 2. (2)

If $X$ is a finite set and $g:X\rightarrow\mathbb{Z}/(p^{n})$ is a function, then $g$ is fibrous iff

[TABLE]

Proof.

(1) Suppose $\sum_{k<p^{n}}b_{k}\zeta^{k}=0$ for some $b_{k}\in\mathbb{Q}$. Then $r(x)=\sum_{k<p^{n}}b_{k}x^{k}\in\mathbb{Q}[x]$ is a polynomial with degree $<p^{n}$ such that $r(\zeta)=0$. So there is some $s(x)\in\mathbb{Q}[x]$ such that $r(x)=s(x)\Phi_{p^{n}}(x)$ where

[TABLE]

is the $p^{n}$th cyclotomic polynomial—the minimal polynomial of $\zeta$ over $\mathbb{Q}$. Since $\deg\Phi_{p^{n}}(x)=p^{n}-p^{n-1}$, it follows that $\deg s(x)<p^{n-1}$. Let

[TABLE]

for some $a_{i}\in\mathbb{Q}$. Expanding $s(x)\Phi_{p^{n}}(x)$ we have

[TABLE]

Comparing coefficients yields

[TABLE]

which is to say, $b_{k}$ depends only $i\equiv k\bmod p^{n-1}$.

(2) Suppose $g$ is fibrous. For each $i<p^{n-1}$, let $a_{i}=|g^{-1}(i)|$. Then

[TABLE]

Conversely, for each $k<p^{n}$ let $c_{k}=|g^{-1}(k)|$. Then

[TABLE]

and (1) implies $c_{k}$ depends only on $k\bmod p^{n-1}$. Hence $g$ is fibrous. ∎

Theorem 12 establishes the equivalence between $q$-BH matrices of dimension $d$ and $\delta$-fibrous functions $f:\mathbb{Z}/(d)\rightarrow\mathbb{Z}/(q)$.

Theorem 12**.**

Let $q=p^{n}$ be a prime power. There is a correspondence between circulant $q$-Butson Hadamard matrices $H$ of dimension $d$ and $\delta$-fibrous functions $f:\mathbb{Z}/(d)\to\mathbb{Z}/(q)$ given by

[TABLE]

Proof.

Suppose $f$ is $\delta$-fibrous. Define the matrix $H=(H_{ij})$ by $H_{ij}=\zeta^{f(i-j)}$ where $\zeta$ is a primitive $q$th root of unity. $H$ is plainly circulant and has all entries $q$th roots of unity. It remains to show that $HH^{*}=dI_{d}$, which is to say that the inner product $r_{j,j+k}$ of column $j$ and column $j+k$ is 0 for each $j$ and each $k\not\equiv 0\bmod d$. For each $k\not\equiv 0\bmod d$ the function $\delta_{k}(x)=f(x+k)-f(x)$ is fibrous. Then we compute

[TABLE]

where the last equality follows from Lemma 11 (2).

Conversely, suppose $H=(H_{i,j})$ is a circulant $q$-Butson Hadammard matrix. Then $H_{i,j}=\zeta^{f(i-j)}$ for some function $f:\mathbb{Z}/(d)\rightarrow\mathbb{Z}/(q)$. Since $HH^{*}=dI_{d}$ we have for each $k\not\equiv 0\bmod d$,

[TABLE]

Lemma 11 (2) then implies $\delta_{k}(x)=f(x+k)-f(x)$ is fibrous. Therefore $f$ is $\delta$-fibrous. ∎

Lemma 13 checks that affine functions are fibrous. We use this in our proof of Theorem 14.

Lemma 13**.**

If $q=p^{n}$ is a prime power, then for all $a\not\equiv 0\bmod q$ and arbitrary $b$, the function $f(x)=ax+b$ is fibrous.

Proof.

Since $a\not\equiv 0\bmod q$,

[TABLE]

Thus, by Lemma 11 (2) we conclude that $f$ is fibrous. ∎

Theorem 14**.**

If $q=p^{n}$ is a prime power, then there exists a circulant $q$-Butson Hadamard matrix of dimension $d=p^{m+n}=p^{m}q$ for each $m\leq n$ unless $(m,n,p)=(0,1,2)$.

Our construction in the proof of Theorem 14 misses the family $(m,n,p)=(0,n,2)$ for each $n\geq 1$. Lemma 15 records a quick observation that circumvents this issue for $n>1$, as our construction does give circulant $2^{n-1}$-BH matrices of dimension $2^{n}$.

Lemma 15**.**

If $q=p^{n}$ is a prime power and there exists a circulant $q$-BH matrix of dimension $d$, then there exists a circulant $p^{k}q$-BH matrix of dimension $d$ for all $k\geq 0$.

Proof.

Every $q$th root of unity is also a $p^{k}q$th root of unity, hence we may view a circulant $q$-BH matrix $H$ of dimension $d$ as a circulant $p^{k}q$-BH for all $k\geq 0$. ∎

Proof of Theorem 14.

Our strategy is to first construct a sequence of functions

[TABLE]

which are fibrous for each $k<p^{m}q$. If $i:\mathbb{Z}/(p^{m})\times\mathbb{Z}/(q)\rightarrow\mathbb{Z}/(p^{m}q)$ is the bijection $i(x,y)=x+p^{m}y$, we define $f:\mathbb{Z}/(p^{m}q)\rightarrow\mathbb{Z}/(q)$ such that $f(z+k)-f(z)=\delta_{k}(x,y)$ when $z=i(x,y)$. Hence $f$ is $\delta$-fibrous and Theorem 12 implies the existence of a corresponding circulant $q$-BH matrix of dimension $p^{m}q$.

Now for each $k\geq 0$ define $\delta_{k}$ by

[TABLE]

where

[TABLE]

Observe that $S_{j}(x)$ counts the integers in the interval $[x,x+j)$ congruent to $-1\bmod p^{m}$ (which only depends on $x\bmod p^{m}$.) Any interval of length $p^{m}j_{1}$ contains precisely $j_{1}$ integers congruent to $-1\bmod p^{m}$. For each $j<p^{m}q$, write $j=j_{0}+p^{m}j_{1}$ with $j_{0}<p^{m}$ and $j_{1}<q$, then

[TABLE]

We show that $\delta_{k}$ is fibrous when $k<p^{m}q$. If $k\not\equiv 0\bmod q$, then for each $x=x_{0}$ the function $\delta_{k}(x_{0},y)$ is affine hence fibrous by Lemma 13. So $\delta_{k}(x,y)$ is fibrous. Now suppose $k=k^{\prime}q$ for some $k^{\prime}<p^{m}$. Using (5) we reduce (4) to

[TABLE]

where $k^{\prime}q=p^{m}(k^{\prime}p^{n-m})=p^{m}\ell$. Here we use our assumption $m\leq n$. The definition of $\chi$ implies

[TABLE]

whence $\delta_{k}(x,y)=\ell x+p^{m}\tbinom{\ell}{2}$. Since $k^{\prime}<p^{m}$ it follows that $\ell=k^{\prime}p^{n-m}<p^{n}=q$, so $\delta_{k}(x,y)$ is affine hence fibrous by Lemma 13.

Define $f:\mathbb{Z}/(p^{m}q)\rightarrow\mathbb{Z}/(q)$ by $f(k)=\delta_{k}(0,0)$. For this to be well-defined, it suffices to check that $\delta_{k+p^{m}q}(x,y)=\delta_{k}(x,y)$ with arbitrary $k$. By (4),

[TABLE]

The argument leading to (6) gives

[TABLE]

Finally, $p^{m}\binom{q}{2}\equiv 0\bmod q$ unless $(m,n,p)=(0,n,2)$. Lemma 15 implies that constructing an example for $(1,n-1,2)$ implies the existence of example for $(0,n,2)$, hence we proceed under the assumption that either $p\neq 2$ or $p=2$ and $m>0$. The case $(m,n,p)=(0,1,2)$ is an exception as one can check explicitly that there are no 2-dimensional Hadamard matrices. Hence it follows that $f$ is well-defined. Let $i:\mathbb{Z}/(p^{m})\times\mathbb{Z}/(q)\rightarrow\mathbb{Z}/(p^{m}q)$ be the bijection $i(x,y)=x+p^{m}y$. To finish the construction we suppose $z=i(x,y)$, show

[TABLE]

and then our proof that $\delta_{k}$ is fibrous for all $k<p^{m}q$ implies $f$ is $\delta$-fibrous. Theorem 12 translates this into the existence of a circulant $q$-BH matrix of dimension $p^{m+n}$. Now,

[TABLE]

∎

Remark*.*

Padraig Cathain brought the work of de Launey [4] to our attention after reading an initial draft. There one finds a construction of circulant $q$-Butson Hadamard matrices of dimension $q^{2}$ for all prime powers $q$ which appears to be closely related to our construction in Theorem 14.

Example 16**.**

We provide two low dimensional examples to illustrate our construction. First we have an 8 dimensional circulant $4$-BH matrix.

[TABLE]

Let $\omega$ be a primitive 3rd root of unity. The following is a 9 dimensional circulant $3$-BH matrix.

[TABLE]

Corollary 17 is an immediate consequence of our main results by Theorem 12.

Corollary 17**.**

If $q=p^{n}$ is a prime power and $d=p^{m+n}$, then there exists a $\delta$-fibrous function $f:\mathbb{Z}/(p^{m+n})\rightarrow\mathbb{Z}/(q)$ iff $m\leq n$, with the one exception of $(m,n,p)=(0,1,2)$.

Closing remarks

Our analysis focused entirely on the existence of circulant $p^{n}$-BH matrices with dimension $d$ a power of $p$. The number theoretic method of Theorem 9 cannot be immediately adapted to the case where $d$ is not a power of $p$, although as we noted earlier, it may be possible to get some constraint with a closer analysis of the eigenvalues of a circulant matrix and the ramification over the primes dividing $d$ in the $d$th cyclotomic extension $\mathbb{Q}(\zeta)/\mathbb{Q}$.

The family of examples constructed in Theorem 14 was found empirically. It would be interesting to know if the construction extends to any dimensions which are not powers of $p$.