Optimal $q$-Ary Error Correcting/All Unidirectional Error Detecting Codes

TL;DR

This paper introduces combinatorial constructions for optimal q-ary codes that correct symmetric errors and detect unidirectional errors, determining exact code lengths for various parameters.

Contribution

It presents new combinatorial methods using graph factorizations and code concatenations to construct optimal error-correcting/detecting codes, expanding known code length results.

Findings

Exact values of shortest code lengths for new parameter families

New combinatorial constructions for q-ary t-EC-AUED codes

Extended the range of known optimal code parameters

Abstract

Codes that can correct up to $t$ symmetric errors and detect all unidirectional errors, known as $t$-EC-AUED codes, are studied in this paper. Given positive integers $q$, $a$ and $t$, let $n_q(a,t+1)$ denote the length of the shortest $q$-ary $t$-EC-AUED code of size $a$. We introduce combinatorial constructions for $q$-ary $t$-EC-AUED codes via one-factorizations of complete graphs, and concatenation of MDS codes and codes from resolvable set systems. Consequently, we determine the exact values of $n_q(a,t+1)$ for several new infinite families of $q,a$ and $t$.

Tables

Table 1. TABLE I: Values of n q ​ ( a , T ) subscript 𝑛 𝑞 𝑎 𝑇 n_{q}(a,T) from Lemmas  IV.3 and IV.4

Table 2. TABLE II: Equivalent Objects for Binary Codes

Equations

$$n_q(sn,T)=2\lambda s(n-1)/(\alpha -1)$$

$$n_q(\alpha q,T)=2\lambda (\alpha q-1)/(\alpha -1)-2$$

$$n_2(a,T)\ge \lceil (4-\frac{2}{\lceil a/2\rceil })T\rceil .$$

$$GB{T}_q(a,T)=\lceil \frac{2a(a-1)T}{a(a-\alpha )-(a-\alpha q)(\alpha +1)}\rceil$$

$$GB{T}_q(a,T_1)+GB{T}_q(a,T_2)=GB{T}_q(a,T_1+T_2),$$

$$n_q(a,T_1+T_2)=GB{T}_q(a,T_1+T_2).$$

$$T_j=\{\{t+j,-t+j\}:1\le t\le k-1\},$$

$$i_1=-y+j_0+(x_0+y)$$

$$i_2=y+j_0+(-x_0-y)$$

$$\begin{array}[]{ccc}0&1&1\\ 1&0&1\\ 1&1&0\\ \end{array}$$

$$\begin{array}[]{ccccc}0&1&2&2&1\\ 1&0&1&2&2\\ 2&1&0&1&2\\ 2&2&1&0&1\\ 1&2&2&1&0\\ \end{array}$$

$$\begin{array}[]{ccccc}1&1&1&1&1\\ 1&2&0&0&2\\ 2&1&2&0&0\\ 0&2&1&2&0\\ 0&0&2&1&2\\ 2&0&0&2&1\\ \end{array}$$

$$\begin{array}[]{lll}\{0,1,2\},&\{3,6\},&\{4,5\}\\ \{0,2,4\},&\{5,6\},&\{1,3\}\\ \{0,1,4\},&\{3,5\},&\{2,6\}\end{array}$$

$$C_4=\left[\begin{array}{c}10022021012\\ 00202112021\\ 20100211202\\ 01020202211\\ 12001122200\\ 11111111111\\ 21221100002\\ 02112002102\\ 12210201020\\ 21012010220\\ 22200020111\\ 00121220120\end{array}\right]$$

$$\displaystyle C_{2}=\begin{bmatrix}211002\\ 202011\\ 201120\\ 100221\\ 112020\\ 120012\\ 010122\\ 021021\\ 022110\\ 012201\\ 102102\\ 111111\\ 121200\\ 210210\\ 220101\\ 001212\end{bmatrix},\$$

$$C_5=\left[\begin{array}{c}01212201221000\\ 22002120022001\\ 01000222102220\\ 10201101112211\\ 11021100201122\\ 21112010101211\\ 20022021210110\\ 22220002000202\\ 12120111120020\\ 10110222011012\\ 02211210010121\\ 00101012222102\end{array}\right]$$

$$\begin{array}[]{lll}\{0,1,2,3\},&\{4,6,8\},&\{5,7,9\}\\ \{0,1,4,5\},&\{2,6,9\},&\{3,7,8\}\\ \{0,2,3,7\},&\{1,6,9\},&\{4,5,8\}\end{array}$$

$$B(n,q)=\left\{(x_1,x_2,\dots ,x_n)\in {\mathbb{F}}_q^n|\sum _{i=1}^n{x}_i=\lceil \frac{n(q-1)}{2}\rceil \right\}$$

Full text

I Introduction

Classical error control codes have been designed for use on binary symmetric channels, i.e., both $1\rightarrow 0$ and $0\rightarrow 1$ errors can occur during transmission. However, errors in some VLSI and optical systems are asymmetric in nature [1, 2], where the error probability from $1$ to [math] is significantly higher than that from [math] to $1$. Practically we can assume that only one type of errors can occur in those systems. These errors are called asymmetric errors.

Different from asymmetric errors, unidirectional errors can be caused by certain faults in digital devices, where both $1\rightarrow 0$ and $0\rightarrow 1$ type of errors are possible, but in any particular word all the errors are of the same type. Digital units that produce unidirectional errors as a consequence of internal failure are data transmission systems, magnetic recording mass memories, and LSI/VLSI circuits such as ROM memories [3]. The number of random errors caused by these failures is usually limited, while the number of unidirectional errors can be large. For this reason, it is useful to consider codes that are capable of correcting a relatively small number of random errors and detecting any number of unidirectional errors. Considerable attention has been paid to this problem, see for example [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16].

In 1973, Varshamov introduced a $q$-ary asymmetric channel [17], where the inputs and outputs of the channel are sequences over the $q$-ary alphabet $R=\{0,1,\ldots,q-1\}$. If the symbol $i$ is transmitted then the only symbols which the receiver can get are $i,i+1,\ldots,q-1$. We say that the type of this error is increasing. Naturally, we have another type of error which is decreasing. The $q$-ary unidirectional channel is the channel on which all errors within a codeword are of the same type (all increasing or all decreasing). Recent work on $q$-ary unidirectional errors can be found in [5, 18, 19, 20, 21] for example.

In this paper, we study constructions for $q$-ary codes which can correct up to $t$ symmetric errors and detect all unidirectional errors (known as $t$-EC-AUED codes). We are only interested in codes that are optimal when considering the shortest lengths for given sizes. Let $n_{q}(a,t+1)$ denote the length of the shortest $q$-ary $t$-EC-AUED code of size $a$. We introduce several combinatorial constructions for $t$-EC-AUED codes and determine the exact values of $n_{q}(a,t+1)$ for new infinite families of $q,a$ and $t$.

Our main results are as follows:

1. (i)

determining values of $n_{3}(a,t+1)$ for $a\leq 12$ and all $t$; 2. (ii)

for integers $k\geq 2$, $n_{k}(a,k-1)=2k-1$ with $k+1\leq a\leq 2k-1$; if $k$ is odd, then $n_{k}(2k,k-1)=2k-1$; 3. (iii)

for prime powers $q\geq 2$, $n_{q}(a,q)=2q+2$ with $2q-1\leq a\leq q^{2}$; 4. (iv)

given positive integers $s$, $\lambda\geq 1$ and $\alpha\geq 2$,

[TABLE]

for all sufficiently large $n$ satisfying $sn\equiv 0\pmod{\alpha}$ and $\lambda s(n-1)\equiv 0\pmod{\alpha-1}$, where $T=\frac{\lambda s(n-1)}{\alpha-1}-\lambda$ and $q=\frac{sn}{\alpha}$; 5. (v)

given positive integers $\lambda\geq 1$ and $\alpha\geq 2$,

[TABLE]

for all sufficiently large $q$ satisfying $\lambda(\alpha q-1)\equiv 0\pmod{\alpha-1}$, where $T=\frac{\lambda\alpha(q-1)}{\alpha-1}-1$. 6. (vi)

values for several other families of $n_{q}(a,t+1)$ are stated in Table I.

Previously, only values of $n_{2}(a,t+1)$ for $a\leq 14$ and $n_{3}(a,t+1)$ for $a\leq 9$ were known by [5].

Our paper is organized as follows. In Section II, we introduce necessary notation and briefly describe the problem status. Section III gives a construction of optimal EC-AUED codes from near one-factorizations, where result (ii) is obtained. In Section IV, we apply concatenation method to various codes with good Hamming distance to get results (iii)-(vi). In Section V, we improve results in [5] and determine completely $n_{3}(a,t+1)$ for $a\leq 12$ and all $t$, which is our result (i). Finally, a conclusion is given in Section VI.

II Preliminaries

Necessary and sufficient conditions for correcting and detecting errors of each of the three types, symmetric, asymmetric and unidirectional, are known in [22, 23]. To state these conditions, we need some necessary notation.

Let $X$ be a finite set, and $R^{X}$ denote the set of vectors of length $|X|$, where each component of a vector ${\sf u}\in R^{X}$ has value in $R$ and is indexed by an element of $X$, that is, ${\sf u}=({\sf u}_{x})_{x\in X}$, and ${\sf u}_{x}\in R$ for each $x\in X$. For ${\sf x},{\sf y}\in R^{X}$, let $N({\sf x},{\sf y})$ denote the number of positions $i$ where ${\sf x}_{i}>{\sf y}_{i}$. If $N({\sf y},{\sf x})=0$, then the vector ${\sf x}$ is said to cover the vector ${\sf y}$ and we write ${\sf x}\geq{\sf y}$. If ${\sf x}\geq{\sf y}$ or ${\sf y}\geq{\sf x}$ the vectors ${\sf x}$ and ${\sf y}$ are said to be ordered, otherwise they are unordered.

A code is a set $\mathcal{C}\subseteq R^{X}$ for some $X$. The elements of $\mathcal{C}$ are called codewords. A code is called a $t$-EC-AUED code if it is able to correct up to $t$ symmetric errors and detect all unidirectional errors. Clearly a code is [math]-EC-AUED if any pair of codewords are unordered. For general $t$, a characterization of when a code is a $t$-EC-AUED code is known as follows.

Theorem II.1**.**

[24]** A code $\mathcal{C}$ is a $t$-EC-AUED code if and only if $N({\sf x},{\sf y})\geq t+1$ and $N({\sf y},{\sf x})\geq t+1$, for all distinct ${\sf x},{\sf y}\in\mathcal{C}$.

Define the asymmetric distance of two vectors ${\sf x}$ and ${\sf y}$ as $d_{as}({\sf x},{\sf y})=\min\{N({\sf x},{\sf y}),N(y,{\sf x})\}$. Then codes with minimum asymmetric distance $T$ are $(T-1)$-EC-AUED codes. Let $n_{q}(a,T)$ denote the length of the shortest $q$-ary $(T-1)$-EC-AUED code of size $a$. We say that a $q$-ary $(T-1)$-EC-AUED code of length $n_{q}(a,T)$ and size $a$ is optimal.

The lower bound derived by Böinck and van Tilborg [4] for the length of binary $(T-1)$-EC-AUED codes is

[TABLE]

In the same paper, they show that if $n_{2}(a,T)=(4-\frac{2}{\left\lceil a/2\right\rceil})T$ holds, then the code must be a constant weight code; and if further $a\equiv 0\pmod{4}$, then $T$ must be divisible by $a/2$.

For non-binary codes, the lower bound of $n_{q}(a,T)$ was generalized in [5].

Theorem II.2**.**

[5]** $n_{q}(a,T)\geq GBT_{q}(a,T)$, where

[TABLE]

and $\alpha=\left\lfloor a/q\right\rfloor$.

The function $GBT$ has a property that for all $\mu\geq 0$, $GBT_{q}(q\mu+(q-1),T)=GBT_{q}(q\mu+q,T)$. That is, by deleting one codeword from the optimal code of size $q\mu+q$, we obtain an optimal code of size $q\mu+(q-1)$.

Lemma II.1**.**

[5]** If $n_{q}(q\mu+q,T)=GBT_{q}(q\mu+q,T)$, then $n_{q}(q\mu+q-1,T)=GBT_{q}(q\mu+q-1,T)$.

In fact, Lemma II.1 can be extended whenever $n_{q}(a,T)=GBT_{q}(a,T)$ and $GBT_{q}(a^{\prime},T)=GBT_{q}(a,T)$, for $a^{\prime}<a$.

Both the Böinck-van Tilborg bound and $GBT$ bound are closely related to Plotkin bound, where the codes achieve the bounds when each symbol occurs almost the same number of times in a fixed position. In such cases, concatenating short codes is a very useful method to construct optimal long codes [25]. As stated in the following lemma, Naydenova and Kløve [5] showed that optimal $t$-EC-AUED codes could be obtained by concatenating two optimal short codes for fixed $q$, $a$ and general $T$.

Lemma II.2**.**

[5]** If $n_{q}(a,T_{1})=GBT_{q}(a,T_{1})$, $n_{q}(a,T_{2})=GBT_{q}(a,T_{2})$, and

[TABLE]

then

[TABLE]

By Theorem II.2, we have $n_{q}(a,T)\geq 2T$ if $q\geq a$. In fact, $n_{q}(a,T)=2T$ in this case since the $a\times 2T$ array formed by $T$ column vectors $(1,2,\ldots,a)$ and $T$ column vectors $(a,a-1,\ldots,1)$ is an optimal code. From now on, we assume that $q<a$.

In [5], the values of $n_{2}(a,T)$ for $a\leq 14$ and all $T$ have been determined by direct constructions and the Böinck-van Tilborg bound. For ternary case, they constructed some optimal codes up to size $9$. We summarize their results for ternary codes as below.

Lemma II.3**.**

[5]** For $T\geq 1$, we have

1. (i)

$n_{3}(a,T)=GBT_{3}(a,T)$* for $a\in\{4,5,6\}$;* 2. (ii)

$\left\lceil 21T/8\right\rceil\leq n_{3}(7,T)\leq\left\lceil 8T/3\right\rceil$; 3. (iii)

$n_{3}(a,T)=\left\lceil 8T/3\right\rceil=GBT_{3}(a,T)$* for $a\in\{8,9\}$ and $T\not\equiv 1\pmod{3}$.*

III A Construction from One-Factorizations

In this section, we give a construction of optimal $q$-ary $t$-EC-AUED codes based on one-factorizations, which yield our main result (ii) by the extension of Lemma II.1. For integers $m\leq n$, the set of integers $\{m,m+1,\ldots,n\}$ is denoted by $[m,n]$. When $m=1$, the set $[1,n]$ is further abbreviated to $[n]$.

Let $a=2k-1$. The ring ${\mathbb{Z}}/a{\mathbb{Z}}$ is denoted by ${\mathbb{Z}}_{a}$. Let $K_{a}$ be a complete graph with vertex set ${\mathbb{Z}}_{a}$. For each $j\in{\mathbb{Z}}_{a}$, take

[TABLE]

where the addition is in ${\mathbb{Z}}_{a}$. Then $\{T_{j}:j\in{\mathbb{Z}}_{a}\}$ is a near one-factorization of $K_{a}$. Each $T_{j}$ is a near one-factor which misses the vertex $j$.

Construction III.1**.**

For each $k\geq 2$, construct a $(2k-1)\times(2k-1)$ array $A$ over $[0,k-1]$, where rows and columns are indexed by ${\mathbb{Z}}_{2k-1}$. For a cell in the $i$th row and the $j$th column, let $A_{i,j}=0$ if $i=j$ and $A_{i,j}=x$ if $i\in\{x+j,-x+j\}$. Let $\mathcal{A}$ be the collection of rows of $A$.

Theorem III.1**.**

$n_{k}(2k-1,k-1)=2k-1$* for all integers $k\geq 2$.*

Proof.

By Theorem II.2, we have $n_{k}(2k-1,k-1)\geq 2k-1$. It suffices to prove that the code $\mathcal{A}$ constructed in Construction III.1 has minimum asymmetric distance $k-1$.

For any two rows ${\sf x}$ and ${\sf y}$ of $A$ indexed by $i_{1}$ and $i_{2}$ respectively, we claim that $N({\sf x},{\sf y})\geq k-1$. In fact, by the definition of near one-factorization, there exists a column indexed by $j_{0}$ and an element $x_{0}\in[k-1]$ such that $\{i_{1},i_{2}\}=\{x_{0}+j_{0},-x_{0}+j_{0}\}\in T_{j_{0}}$, i.e., $A_{i_{1},j_{0}}=A_{i_{2},j_{0}}=x_{0}$. Without loss of generality, assume that $i_{1}=x_{0}+j_{0}$ and $i_{2}=-x_{0}+j_{0}$. Thus for each $y\in[k-1]$, we have

[TABLE]

and

[TABLE]

in ${\mathbb{Z}}_{2k-1}$. By the construction of $A$, we have $A_{i_{1},j_{0}-y}=A_{i_{2},j_{0}+y}$ which equals $x_{0}+y$ or $-(x_{0}+y)$ whoever falls in $[k-1]$. So for each $y\in[k-1]$, if $A_{i_{1},j_{0}-y}>A_{i_{2},j_{0}-y}$, we must have $A_{i_{1},j_{0}+y}<A_{i_{2},j_{0}+y}$. Thus $N({\sf x},{\sf y})=N({\sf y},{\sf x})=k-1$. ∎

Example III.1**.**

Let $k=2$ and $k=3$. Then applying Construction III.1 and Theorem III.1 gives an optimal binary [math]-EC AUED codes of size three, and an optimal ternary $1$-EC AUED codes of size five.

[TABLE]

[TABLE]

Remark 1: The array $A$ from Construction III.1 has extra property that in each row, each nonzero element occurs twice and the zero element occurs exactly once.

Construction III.2**.**

For odd integers $k\geq 3$, let $B$ be a $(2k-1)\times(2k-1)$ array with the entry $B_{i,j}\equiv A_{i,j}+(k-1)/2\pmod{k}$, where $A$ is the array from Construction III.1. Let ${\sf u}$ be a vector of length $2k-1$ with all entries being $(k-1)/2$. Denote $\mathcal{B}$ the collection of rows of $B$ and let $\mathcal{B}^{\prime}=\mathcal{B}\cup\{{\sf u}\}$.

Theorem III.2**.**

$n_{k}(2k,k-1)=2k-1$* for all odd integers $k\geq 3$.*

Proof.

The lower bound can be checked by Theorem II.2. For the upper bound, we only need to show that $\mathcal{B}$ in Construction III.2 is a $(k-2)$-EC-AUED code by Remark 1. This follows from the fact that $B_{i_{1},j_{0}-y}=B_{i_{2},j_{0}+y}$ for each $y\in[1,k-1]$ as in the proof of Theorem III.1. ∎

Example III.2**.**

Applying Construction III.2 and Theorem III.2 with $k=3$ gives an optimal ternary $1$-EC AUED codes of size six.

[TABLE]

IV A Construction by Concatenation

We first give a simple but very useful construction of EC-AUED codes by concatenation. As mentioned in Section II, this method has been widely used to construct codes achieving Plotkin type bounds. For any $q$-ary word $c=(c_{1},c_{2},\ldots,c_{n})$, let $q-1-c:=(q-1-c_{1},q-1-c_{2},\ldots,q-1-c_{n})$ and $c|(q-1-c)=(c_{1},\ldots,c_{n},q-1-c_{1},\ldots,q-1-c_{n})$. For any two words ${\sf x}$ and ${\sf y}$, the Hamming distance of ${\sf x}$ and ${\sf y}$, denoted by $d_{H}({\sf x},{\sf y})$, is the number of positions $i$ such that ${\sf x}_{i}={\sf y}_{i}$. It is obvious that $d_{H}({\sf x},{\sf y})=N({\sf x},{\sf y})+N({\sf y},{\sf x})$.

Lemma IV.1**.**

Let $C$ be a $q$-ary code of length $n$ with minimum Hamming distance $d$. Then $\{c|(q-1-c):c\in C\}$ is a $q$-ary $(d-1)$-EC-AUED code of length $2n$ with $|C|$ words.

Proof.

For any two words ${\sf x}=c|(q-1-c)$ and ${\sf y}=c^{\prime}|(q-1-c^{\prime})$, since $N(q-1-c,q-1-c^{\prime})=N(c^{\prime},c)$, we have $N({\sf x},{\sf y})=N(c,c^{\prime})+N(q-1-c,q-1-c^{\prime})=N(c,c^{\prime})+N(c^{\prime},c)=d_{H}(c,c^{\prime})\geq d$. It’s similar that $N({\sf y},{\sf x})\geq d$. ∎

By Lemma IV.1, we can construct good EC-AUED codes from codes with large Hamming distance.

Theorem IV.1**.**

$n_{q}(q^{2},q)=2q+2$* for all prime powers $q$.*

Proof.

The lower bound is checked by Theorem II.2. The upper bound is obtained by applying Lemma IV.1 to $q$-ary MDS codes of length $q+1$ and size $q^{2}$ with minimum Hamming distance $q$ [26, Chapter 5]. ∎

IV-A Constructions from Set Systems

A set system is a pair ${\mathfrak{S}}=(X,\mathcal{A})$, where $X$ is a finite set of points and $\mathcal{A}\subseteq 2^{X}$. Elements of $\mathcal{A}$ are called blocks. The order of $\mathfrak{S}$ is the number of points in $X$, and the size of $\mathfrak{S}$ is the number of blocks in $\mathcal{A}$. Let $K$ be a set of positive integers. A set system $(X,\mathcal{A})$ is $K$-uniform if $|A|\in K$ for all $A\in\mathcal{A}$. A parallel class of a set system $(X,\mathcal{A})$ is a set ${\cal P}\subseteq\mathcal{A}$ that partitions $X$. A resolvable set system is a set system whose set of blocks can be partitioned into parallel classes. We refer the readers to [27] for other related concepts in combinatorial design theory.

Definition IV.1**.**

Let $(X,\mathcal{A})$ be a $\{k\}$-uniform set system of order $n$. Then it is an $(n,k,\lambda)$-packing if each pair of $X$ occurs in at most $\lambda$ blocks of $\mathcal{A}$.

Given a resolvable $(qk,k,\lambda)$-packing of $n$ parallel classes, arbitrarily order the $q$ blocks in each parallel class by elements in $[0,q-1]$. Define an $qk\times n$ $q$-ary matrix $A$ by indexing each column by a parallel class and each row by a point of the packing. For each parallel class, the corresponding column has the symbol $i$ in the rows indexed by the points in the $i$th block. Since each pair of points occurs in at most $\lambda$ blocks, the rows of $A$ form a $q$-ary code of Hamming distance at least $n-\lambda$. Note that this correspondence is the one used by Semakov and Zinoviev [28] to show the equivalence between equidistant codes and RBIBDs. Recently, this method is used again to construct optimal equitable symbol weight codes, see for example [29, 30]. By applying Lemma IV.1 to this equivalence, we have the following result.

Lemma IV.2**.**

Suppose that there exists a resolvable $(a,k,\lambda)$-packing, which has $n$ parallel classes each consisting of $q$ blocks, $q=a/k$. Then there exists a $q$-ary $t$-EC-AUED code of size $a$ and length $2n$ with $t=n-\lambda-1$.

Next, we apply Lemma IV.2 to some concrete combinatorial objects to determine the values of $n_{q}(a,T)$.

Definition IV.2**.**

Let $(X,\mathcal{A})$ be a $\{k\}$-uniform set system and let $\mathcal{G}$ be a partition of $X$ into subsets, called groups. The triple $(X,\mathcal{G},\mathcal{A})$ is a group divisible design (GDD) when every 2-subset of $X$ not contained in a group is contained in exactly $\lambda$ block, and $|A\cap G|\leq 1$ for all $A\in\mathcal{A}$ and $G\in\mathcal{G}$.

We denote such a GDD $(X,\mathcal{G},\mathcal{A})$ by $(k,\lambda)$-GDD. It is obvious that a $(k,\lambda)$-GDD $(X,\mathcal{G},\mathcal{A})$ is an $(n,k,\lambda)$-packing with $n=|X|$. The type of a GDD $(X,\mathcal{G},\mathcal{A})$ is the multiset $\langle|G|:G\in\mathcal{G}\rangle$. When more convenient, the exponential notation is used to describe the type of a GDD: a GDD of type $g_{1}^{t_{1}}g_{2}^{t_{2}}\cdots g_{s}^{t_{s}}$ is a GDD where there are exactly $t_{i}$ groups of size $g_{i}$, $i\in[s]$. When a GDD is resolvable, we denote it by RGDD. A $(k,\lambda)$-GDD of type $1^{n}$ is called a balanced incomplete block design, denoted by BIBD$(n,k,\lambda)$ and RBIBD$(n,k,\lambda)$ when it is resolvable.

Theorem IV.2**.**

[31]** Fix integers $g,\lambda\geq 1$ and $k\geq 2$. There exists an integer $u_{0}(g,k)$ such that for all $u\geq u_{0}$, a $(k,\lambda)$-RGDD of type $g^{u}$ exists if and only if $\lambda g(u-1)\equiv 0\pmod{k-1}$ and $gu\equiv 0\pmod{k}$.

Lemma IV.3**.**

Suppose that there exists an $(\alpha,\lambda)$-RGDD of type $s^{n}$, such that $2\lambda(s-1)<sn-\alpha$. Then $n_{q}(sn,T)=2\lambda s(n-1)/(\alpha-1)$, where $T=\frac{\lambda s(n-1)}{\alpha-1}-\lambda$ and $q=\frac{sn}{\alpha}$.

Proof.

It is easy to check that when $2\lambda(s-1)<sn-\alpha$, we have $n_{q}(sn,T)\geq 2\lambda s(n-1)/(\alpha-1)$ by Theorem II.2. The equality could be obtained by Lemma IV.2 and the fact that the given GDD is a resolvable $(sn,\alpha,\lambda)$-packing. ∎

Theorem IV.3**.**

Given positive integers $s$, $\lambda\geq 1$ and $\alpha\geq 2$, $n_{q}(sn,T)=2\lambda s(n-1)/(\alpha-1)$ for all sufficiently large $n$ satisfying $sn\equiv 0\pmod{\alpha}$ and $\lambda s(n-1)\equiv 0\pmod{\alpha-1}$, where $T=\frac{\lambda s(n-1)}{\alpha-1}-\lambda$ and $q=\frac{sn}{\alpha}$.

Proof.

For fixed $s,\alpha$ and $\lambda$, we have $2\lambda(s-1)<sn-\alpha$ for sufficiently large $n$. Hence the conclusion follows by Lemma IV.3 and the asymptotic existence of $(\alpha,\lambda)$-RGDD of type $s^{n}$ in Theorem IV.2. ∎

Lemma IV.4**.**

If there exists an RBIBD$(\alpha q,\alpha,\lambda)$ with $q\geq 3$, then $n_{q}(\alpha q,T)=2\lambda(\alpha q-1)/(\alpha-1)-2$, where $T=\frac{\lambda\alpha(q-1)}{\alpha-1}-1$.

Proof.

Delete one parallel class from the RBIBD$(\alpha q,\alpha,\lambda)$ to get a resolvable $(\alpha q,\alpha,\lambda)$-packing. Then apply Lemma IV.2. ∎

Theorem IV.4**.**

Given positive integers $\lambda\geq 1$ and $\alpha\geq 2$, $n_{q}(\alpha q,T)=2\lambda(\alpha q-1)/(\alpha-1)-2$ for all sufficiently large $q$ satisfying $\lambda(\alpha q-1)\equiv 0\pmod{\alpha-1}$, where $T=\frac{\lambda\alpha(q-1)}{\alpha-1}-1$.

Proof.

By the asymptotic existence RBIBD$(\alpha q,\alpha,\lambda)$ in Theorem IV.2. ∎

In Table I, we give some examples of exact values of $n_{q}(a,T)$ determined by Lemmas IV.3, IV.4 and the extension of Lemma II.1. The existence of combinatorial objects used in this table can be found in [27].

Before closing this section, we note that for two binary vectors ${\sf x}$ and ${\sf y}$ of equal number of $1$’s, we have that $N({\sf x},{\sf y})=N({\sf y},{\sf x})=\frac{1}{2}d_{H}({\sf x},{\sf y})$. So a binary constant weight code with minimum Hamming distance $d$ is a $(\frac{d}{2}-1)$-EC AUED code. It’s well known that the rows of the incidence matrix of a BIBD form a binary constant weight code. In Table II, we give two examples of equivalent objects for optimal binary EC AUED codes. However, the existence of corresponding BIBDs used in Table II is very rare by referring to [32].

V Optimal Ternary $t$-EC-AUED Codes

In this section, we give some direct constructions of optimal ternary $t$-EC-AUED codes up to size $12$. For some of the codes we search directly by computer, but when the length becomes big, the searching space will be huge. In this case, we map each ternary code to be a resolvable set system as in Section IV. Suppose there is a ternary $t$-EC-AUED code of size $a$ and length $n$. Let the rows be indexed by $X$ of size $a$, then for each column, we obtain three blocks by collecting all the indices of rows with same entries. Thus we get a resolvable set system of order $a$ and size $3n$, where each parallel class has three blocks. Conversely, we can get the corresponding ternary code from such a resolvable set system. However, to ensure that the code is a $t$-EC-AUED code, the set system must satisfy extra conditions, which are not easy to be characterized.

In the following constructions, if we construct a resolvable set system instead of the ternary code, we list the three blocks in each parallel class in order, for which the entries in the corresponding rows will be assigned to [math], $1$ and $2$ respectively. If we list the optimal code itself, we usually denote $C_{T}$ the optimal $(T-1)$-EC-AUED code.

Further, in design theory, people usually equip the desired designs with some group structures to reduce the search space. What they do is try to find a partial result, which can be developed to the complete desired design by using the group structure. For example, if a block $B$ is developed by ${\mathbb{Z}}_{n}$, then $B_{i}$, $i\in{\mathbb{Z}}_{n}$ are obtained such that $B_{i}=\{b+i:b\in B\}$. We will apply this idea to some of our constructions.

Lemma V.1**.**

$n_{3}(7,8)=21$.

Proof.

Let $X={\mathbb{Z}}_{7}$. The following nine blocks form three parallel classes, where each row is a parallel class. Develop them to $21$ parallel classes by ${\mathbb{Z}}_{7}$ and keep the order of blocks in each parallel class. Then one can check this resolvable set system gives a ternary $7$-EC-AUED code of size $7$ and length $21$, which is optimal by the fact that $GBT_{3}(7,8)=21$. In fact, the codewords are the rows of the $7\times 21$ matrix $M=(A|B|C)$ where $A$, $B$ and $C$ are circulant matrices with rows indexed by ${\mathbb{Z}}_{7}$. The leftmost column of $A$ has zeroes in rows $0,1,2$, ones in rows $3$ and $6$, and twos in rows $4$ and $5$; the leftmost column of $B$ has zeroes in rows $0,2,4$, ones in rows $5$ and $6$, and twos in rows $1$ and $3$; and the leftmost column of $C$ has zeroes in rows $0,1,4$, ones in rows $3$ and $5$, and twos in rows $2$ and $6$.

[TABLE]

In fact, this resolvable set system can be found in [33, Example 2.3] as a class-uniformly resolvable design with partition $2^{2}3^{1}$.

∎

Theorem V.1**.**

$n_{3}(7,T)=\left\lceil\frac{21}{8}T\right\rceil$* for all $T\geq 1$.*

Proof.

For $a=7$, we have $GBT_{3}(7,T)=\left\lceil\frac{21}{8}T\right\rceil$. When $1\leq T\leq 7$, $n_{3}(7,T)=GBT_{3}(7,T)$ is known by [5]. By Lemma V.1, $n_{3}(7,8)=GBT_{3}(7,8)$. Hence the construction $C_{T}=C_{8}|C_{T-8}$ gives the optimal code of length $\left\lceil\frac{21}{8}T\right\rceil$ for all $T$ by Lemma II.2. ∎

In [5], the authors stated that $n_{3}(9,1)=4=GBT_{3}(9,1)+1$ and $n_{3}(9,4)=12=GBT_{3}(9,4)+1$ by computer search. However, the latter is not true. In fact, we find a $3$-EC-AUED code of length $11$ and with bigger size $12$, which meets the bound in Theorem II.2.

Lemma V.2**.**

$n_{3}(a,4)=11$* for $8\leq a\leq 12$.*

Proof.

For $8\leq a\leq 12$, we have $GBT_{3}(a,4)=11$. A $3$-EC-AUED code of size $12$ and length $11$ is constructed below. For $8\leq a\leq 11$, the optimal codes are obtained by collecting any set of $a$ codewords from $C_{4}$.

[TABLE]

∎

Hence by Lemmas II.2 and II.3, we determine all values of $n_{3}(8,T)$ and $n_{3}(9,T)$.

Theorem V.2**.**

$n_{3}(8,T)=n_{3}(9,T)=\left\lceil\frac{8}{3}T\right\rceil$* for all $T>1$.*

Next, we study values of $n_{3}(a,T)$ for $a\in\{10,11,12\}$. By Lemma II.1, it is enough to consider the cases $a=10,12$, for which we have $GBT_{3}(10,T)=\left\lceil\frac{30T}{11}\right\rceil$ and $GBT_{3}(12,T)=\left\lceil\frac{11T}{4}\right\rceil$.

Lemma V.3**.**

$n_{3}(a,2)=6$* for $7\leq a\leq 16$ and $n_{3}(a,3)=9$ for $10\leq a\leq 25$.*

Proof.

For $7\leq a\leq 16$, we have $GBT_{3}(a,2)=6$, while for $10\leq a\leq 25$, we have $GBT_{3}(a,3)=9$. An optimal $1$-EC-AUED code of size $16$ and a $2$-EC-AUED code of size $25$ are listed below. Optimal codes with smaller sizes can be obtained by deleting some codewords from $C_{2}$ and $C_{3}$ respectively.

[TABLE]

∎

By now, we have determined $n_{3}(12,T)$ for $2\leq T\leq 4$. Since $n_{3}(9,1)=4$, we have $n_{3}(12,1)\geq 4$ which is bigger than $GBT_{3}(12,1)=3$. By Lemma II.2, we still need to determine $n_{3}(12,5)$ to construct all optimal codes of size $12$.

Lemma V.4**.**

$n_{3}(a,5)=14$* for $10\leq a\leq 12$.*

Proof.

For $10\leq a\leq 12$, we have $GBT_{3}(a,5)=14$. An optimal code of size $12$ is given below.

[TABLE]

∎

Hence by Lemmas II.2 and II.3, we determine all values of $n_{3}(12,T)$.

Theorem V.3**.**

$n_{3}(11,T)=n_{3}(12,T)=\left\lceil\frac{11T}{4}\right\rceil$* for all $T>1$.*

By simple computation, we know that $GBT_{3}(10,T)=GBT_{3}(12,T)$ for most integers $T$. The smallest integer $T$ with $GBT_{3}(10,T)<GBT_{3}(12,T)$ is $11$.

Lemma V.5**.**

$n_{3}(10,11)=30$.

Proof.

Let $X={\mathbb{Z}}_{10}$. The following nine blocks form three parallel classes in each row. Develop them to $30$ parallel classes by ${\mathbb{Z}}_{10}$ and keep the order of blocks in each parallel class. Then one can check this resolvable set system gives a ternary $10$-EC-AUED code of size $10$ and length $30$, which is optimal by Theorem II.2. In fact, the codewords are the rows of the $10\times 30$ matrix $M=(A|B|C)$ where $A$, $B$ and $C$ are circulant matrices with rows indexed by ${\mathbb{Z}}_{10}$. The leftmost column of $A$ has zeroes in rows $0,1,2,3$, ones in rows $4,6,8$, and twos in rows $5,7,9$; the leftmost column of $B$ has zeroes in rows $0,1,4,5$, ones in rows $2,6,9$, and twos in rows $3,7,8$; and the leftmost column of $C$ has zeroes in rows $0,2,3,7$, ones in rows $1,6,9$, and twos in rows $4,5,8$.

[TABLE]

∎

Now we are in a position to determine $n_{3}(10,T)$ for all $T>1$.

Theorem V.4**.**

$n_{3}(10,T)=\left\lceil\frac{30T}{11}\right\rceil$* for all $T>1$.*

Proof.

For $a=10$, we have $GBT_{3}(10,T)=\left\lceil\frac{30T}{11}\right\rceil$. For $T=2,\ldots,10,12$, the bound is met since $GBT_{3}(10,T)=GBT_{3}(12,T)$ for these cases. For $T=11$, the bound is achieved by Lemma V.5. The optimal code $C_{T}$ for all $T\geq 13$ is given by recursion $C_{T}=C_{11}|C_{T-11}$. ∎

Finally, for completeness, when $T=1$, we note that $n_{3}(a,1)=4=GBT_{3}(a,1)+1$ for $8\leq a\leq 19$ since by de Bruijn et al. [34],

[TABLE]

is a [math]-EC-AUED code with maximal size for given length $n$.

VI Conclusion

We investigated the length of the shortest $q$-ary $t$-EC-AUED codes of size $a$. A direct construction of optimal codes was given via one-factorizations of complete graphs. We further provided a general construction of a $(d-1)$-EC-AUED code of length $2n$ from a code of length $n$ and minimum Hamming distance $d$. Finally, we would like to suggest the study of codes for which the words are the rows of a concatenation of circulant matrices, similar to those constructed from resolvable packings in Section V.

Acknowledgments

The authors thank the anonymous reviewers and the associate editor for their constructive comments and suggestions that greatly improved the quality of this paper.