Double Circulant Self-dual Codes on Sextic Cyclotomy

Tongjiang Yan^a, Tao Wang^a, Wenpeng Gao^a, Xvbo Zhao^a
^a College of Science, China University of Petroleum,
Qingdao 266555, Shandong, China
Corresponding author: Tongjiang Yan
Email: [email protected]; [email protected];
[email protected]; zhaoxubo

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[email protected]

Abstract

This paper contributes to construct double circulant self-dual codes by sextic cyclotomy. Generator matrixes of a family of pure double circulant codes and a family of double circulant codes with boundary are formed from sextic cyclotomic classes. These codes are proved to be self-dual on certain conditions. Moreover, experiments show that some of them are optimal self-dual codes or best than ever codes over $\mathrm{GF}(2)$ and $\mathrm{GF}(4)$ .

Index Terms. Self-dual code, double circulant code, sextic cyclotomy.

1 Introduction

An $[n,k,d]$ linear code $C$ over the Galois field $\mathrm{GF}(q)$ is a $k$ -dimensional subspace of $\mathrm{GF}(q)^{n}$ with the minimum Hamming weight $d$ , where $q$ is a prime power. Its generator matrix $G$ is a $k\times n$ matrix over $\mathrm{GF}(q)$ whose rows span the code. Every element of $C$ is called a codeword. The Euclidean inner product between two codewords $x=(x_{0},x_{1},\dots,x_{n-1})$ and $y=(y_{0},y_{1},\dots,y_{n-1})$ is defined by

$(x,y)=\sum\limits_{i=0}^{n-1}x_{i}y_{i}.$

The orthogonal subspace of $C$ in the linear space $\mathrm{GF}(q)^{n}$ denoted by

$C^{\perp}=\{x\in\mathrm{GF}(q)^{n}|(x,c)=0,\textrm{for all }c\in C\}$

is defined as an Euclidean dual code of $C$ . Moreover, the code $C$ is Euclidean self-dual if $C=C^{\perp}$ , hereinafter referred to as a self-dual code.

Self-dual codes play an important role in classical linear error correcting codes and constructing quantum error correcting codes [6, 8]. But it is difficult to construct them with larger minimum Hamming distances. Although the existence of progressive self-dual code groups has been proved, we are still unable to find out exactly in most cases. A double circulant self-dual code employs one generator matrix consists of two circulant matrices and possesses many good algebraic properties [5, 3]. With more new double circulant self-dual codes were presented, these provide a remarkable way to construct self-dual codes [3, 13, 4, 12, 9, 10, 11]. Recently, by constructting double circulant generator matrixes using cyclotomy of order four modulo an odd prime $p$ , Zhang and Ge presented some double circulant self-dual codes with optimal minimum distances [13]. Next, we will construct several double circulant self-dual codes with optimal minimum distances by sextic cyclotomy modulo $p$ . All computations have been done by MAGMA [2].

The Hamming weight of a codeword, denoted by $wt(x)$ , is defined as the number of nonzero coordinates of $x$ . The minimum nonzero Hamming weight of all codewords in a linear code $C$ is equal to its minimum Hamming distance $d(C)$ .

Lemma 1

[6] Let $C$ be a self-dual code.

(1) For the case $q=2$ ,

$d(C)\leq\begin{cases}4\left\lfloor\displaystyle\frac{n}{24}\right\rfloor+4,&\textrm{$n\not\equiv 22\bmod{24}$,}\vspace{3mm}\\ 4\left\lfloor\displaystyle\frac{n}{24}\right\rfloor+6,&\textrm{$n\equiv 22\bmod{24}$.}\end{cases}$

(2) For the case $q=4$ ,

d(C)\leq 4\left\lfloor\displaystyle\frac{n}{12}\right\rfloor+3.

The self-dual code $C$ is called optimal if and only if it can get the highest possible minimum distance. If the minimum Hamming distance of the self-dual code exceeds those of all the codes previously constructed, we call it best than ever. All the known optimal and best than ever self-dual codes can be found in [7].

This paper is organized as follows. We firstly present some definitions and preliminaries in Section $2$ . In Section $3$ , several families of double circulant self-dual codes are obtained. Section $4$ concludes the paper finally.

2 Primaries

2.1 cyclotomy

Suppose $p=6f+1$ is an odd prime and $f$ is a positive integer. Let $\gamma$ be a fixed primitive element on the Galois field $\mathrm{GF}(p)$ . We define a sextic cyclotomic classes $C_{i}$ as

C_{i}=\{\gamma^{6j+i}:0\leq j\leq\displaystyle\frac{p-1}{6}-1\},

where $0\leq i\leq 5$ . That means $C_{i}=\gamma^{i}C_{0}$ . For arbitrary integers $m,n$ satisfying $0\leq m,n\leq 5$ , a sextic cyclotomic number is define as [13] :

(m,n)=\left|C_{m}+1\cap C_{n}\right|.

The following lemma summarizes some basic properties of the cyclotomic number.

Lemma 2

[1] Let $p=6f+1$ be some odd prime. Then

\begin{split}&(a)\qquad(i,j)=(i^{\prime},j^{\prime}),\;when\;i\equiv i^{\prime}\bmod{6},\;j\equiv j^{\prime}\bmod{6};\\ &(b)\qquad(i,j)=(6-i,j-i)=\left\{\begin{aligned} (j,i),\qquad\quad f\;is\;even;\\ (j+3,i+3),\;f\;is\;odd;\\ \end{aligned}\right.\\ &(c)\qquad\sum\limits_{i=0}^{5}=f-\delta_{j},\;where\;\delta_{j}=\left\{\begin{aligned} 1,&\;if\;j=0\bmod{6};\\ 0,&\;otherwise.\end{aligned}\right.\end{split}

Lemma 3

Suppose $p=12l+7$ is a prime. Then

	$\displaystyle(0,0)=$	$\displaystyle(3,0)=(3,3),\,\,(0,1)=(2,5)=(4,3),\,\,(0,2)=(1,4)=(5,3)\,\,\,(0,4)=(1,3)=(5,2)$
	$\displaystyle(0,5)=$	$\displaystyle(2,3)=(4,1),\,\,(1,0)=(2,2)=(3,4)=(4,0)=(3,1)=(5,5),\,\,\,(2,1)=(4,5)$
	$\displaystyle(1,1)=$	$\displaystyle(2,0)=(3,2)=(3,5)=(4,4)=(5,0),\,\,(1,2)=(1,5)=(2,4)=(4,2)=(5,1)=(5,4).$

$\mathbf{Proof.}$ The proof is straightforward from the Lemma 2. $\blacksquare$

For simplicity, in the following we define

	$\displaystyle A:=$	$\displaystyle(0,0),\qquad B:=(0,1),\qquad C:=(0,2),\qquad D:=(0,3),\qquad E:=(0,4),$
	$\displaystyle F:=$	$\displaystyle(0,5),\qquad G:=(1,0),\qquad H:=(1,1),\qquad I:=(1,2),\qquad J:=(2,1).$

Lemma 4

[1] Suppose $p=12l+7$ is a prime. Let $g$ be a primitive root of $p$ , satisfying $g^{t}\equiv 2\bmod{p}$ , where $t$ is a positive integer. Then there are positive integers $x$ , $y$ , such that $p=x^{2}+3y^{2}$ , and $x\equiv 1\bmod{3}$ , $y\equiv-t\bmod{3}$ . Thus the sextic cyclotomic numbers are
(a) when $t\equiv 0\bmod 3$ ,

	$\displaystyle A$	$\displaystyle=\frac{p-11-8x}{36},\,\,\,\,\,\qquad B=\frac{p+1-2x+12y}{36},\qquad C=\frac{p+1-2x+12y}{36},$
	$\displaystyle D$	$\displaystyle=\frac{p+1+16x}{36},\,\,\,\,\,\qquad E=\frac{p+1-2x-12y}{36},\,\,\,\,\,\,\quad F=\frac{p+1-2x-12y}{36},$
	$\displaystyle G$	$\displaystyle=\frac{p-5+4x+6y}{36},\quad\ H=\frac{p-5+4x-6y}{36},\,\,\qquad I=\frac{p+1-2x}{36},$
	$\displaystyle J$	$\displaystyle=\frac{p+1-2x}{36};$

(b) when $t\equiv 1\bmod 3$ ,

	$\displaystyle A=$	$\displaystyle\frac{p-11-2x}{36},\,\,\,\,\qquad\qquad B=\frac{p+1+4x}{36},\qquad\,\,\,\,\qquad C=\frac{p+1-2x+12y}{36},$
	$\displaystyle D=$	$\displaystyle\frac{p+1+10x-12y}{36},\qquad E=\frac{p+1-8x-12y}{36},\qquad F=\frac{p+1-2x+12y}{36},$
	$\displaystyle G=$	$\displaystyle\frac{p-5-2x+6y}{36},\,\,\,\,\,\qquad H=\frac{p-5+4x-6y}{36},\,\,\,\,\,\qquad I=\frac{p+1+4x}{36},$
	$\displaystyle J=$	$\displaystyle\frac{p+1-8x+12y}{36};$

	$\displaystyle A=$	$\displaystyle\frac{p-11-2x}{36},\,\,\,\qquad\qquad B=\frac{p+1-2x-12y}{36},\,\qquad C=\frac{p+1-8x+12y}{36},$
	$\displaystyle D=$	$\displaystyle\frac{p+1+10x+12y}{36},\qquad E=\frac{p+1-2x-12y}{36},\qquad F=\frac{p+1+4x}{36},$
	$\displaystyle G=$	$\displaystyle\frac{p-5+4x+6y}{36},\,\,\,\,\,\qquad H=\frac{p-5-2x-6y}{36},\,\,\,\,\qquad I=\frac{p+1+4x}{36},$
	$\displaystyle J=$	$\displaystyle\frac{p+1-8x+12y}{36}.$

2.2 Sextic residue double circulant codes

Let $p=6f+1$ be an odd prime, where $f$ is a positive integer. Let $\vec{m}:=(m_{0},m_{1},m_{2},m_{3},m_{4},m_{5},m_{6})\in\mathrm{GF}(q)^{7}$ , where $q$ is a prime power. Now we define $C_{p}(\vec{m})$ as a matrix $R=(r_{ij})_{p\times p}$ on $\mathrm{GF}(q)$ , where $1\leq i,j\leq p$ , and

r_{ij}=\left\{\begin{aligned} m_{0},&\quad\mathrm{if}\quad j=i,\\ m_{s+1},&\quad\mathrm{if}\quad j-i\in C_{s},\end{aligned}\right.

(1)

where $s+1$ means $s+1\bmod{6}$ , and $s=0,1,\cdots,5$ .

Definition 1

Let $P_{p}(R)$ and $B_{p}(\alpha,R)$ be two linear codes with the following generator matrixes

\left(\begin{array}[]{cc}I_{p},R\end{array}\right),\,\,\,\,\,\,\,\,\left(\begin{array}[]{cccccc}&\alpha&1&\cdots&1\\ &-1&&&\\ I_{p+1}&\vdots&&R&\\ &-1&&&\end{array}\right)

respectively, where $\alpha\in\mathrm{GF}(q)$ and $R$ is the cyclic matrix defined in Equation (1). We call $P_{p}(R)$ a sextic residue pure double circulant code and $B_{p}(\alpha,R)$ a sextic residue double circulant code with boundary. In general, we call them sextic residue double circulant codes.

Define $A_{0}$ and $J_{p}$ as the $p\times p$ identity matrix and all-one matrix, respectively. Then $C_{p}(1,0,0,0,0,0,0)=A_{0}$ , $C_{p}(1,1,1,1,1,1,1)=J_{p}$ . Define

$\displaystyle A_{1}:=$	$\displaystyle C_{p}(0,1,0,0,0,0,0)\qquad A_{2}:=C_{p}(0,0,1,0,0,0,0)$	(2)
$\displaystyle A_{3}:=$	$\displaystyle C_{p}(0,0,0,1,0,0,0)\qquad A_{4}:=C_{p}(0,0,0,0,1,0,0)$
$\displaystyle A_{5}:=$	$\displaystyle C_{p}(0,0,0,0,0,1,0)\qquad A_{6}:=C_{p}(0,0,0,0,0,0,1)$

Then we can get the following results.

Lemma 5

Suppose $p$ is a prime and $p=12l+7$ . Then the matrices in Equation (2) has the following properties

	$\displaystyle A_{1}=$	$\displaystyle A_{4}^{T},\qquad A_{2}=A_{5}^{T},\qquad A_{3}=A_{6}^{T},$
	$\displaystyle A_{1}^{2}=$	$\displaystyle AA_{1}+BA_{2}+CA_{3}+DA_{4}+EA_{5}+FA_{6},$
	$\displaystyle A_{2}^{2}=$	$\displaystyle FA_{1}+AA_{2}+BA_{3}+CA_{4}+DA_{5}+EA_{6},$
	$\displaystyle A_{3}^{2}=$	$\displaystyle EA_{1}+FA_{2}+AA_{3}+BA_{4}+CA_{5}+DA_{6},$
	$\displaystyle A_{4}^{2}=$	$\displaystyle DA_{1}+EA_{2}+FA_{3}+AA_{4}+BA_{5}+CA_{6},$
	$\displaystyle A_{5}^{2}=$	$\displaystyle CA_{1}+DA_{2}+EA_{3}+FA_{4}+AA_{5}+BA_{6},$
	$\displaystyle A_{6}^{2}=$	$\displaystyle BA_{1}+CA_{2}+DA_{3}+EA_{4}+FA_{5}+AA_{6},$
	$\displaystyle A_{1}A_{2}=$	$\displaystyle A_{2}A_{1}=GA_{1}+HA_{2}+IA_{3}+EA_{4}+CA_{5}+IA_{6},$
	$\displaystyle A_{1}A_{3}=$	$\displaystyle A_{3}A_{1}=HA_{1}+JA_{2}+GA_{3}+FA_{4}+IA_{5}+BA_{6},$
	$\displaystyle A_{1}A_{4}=$	$\displaystyle A_{4}A_{1}=(2l+1)A_{0}+AA_{1}+GA_{2}+HA_{3}+AA_{4}+GA_{6}+HA_{6},$
	$\displaystyle A_{1}A_{5}=$	$\displaystyle A_{5}A_{1}=GA_{1}+FA_{2}+IA_{3}+BA_{4}+HA_{6}+JA_{6},$
	$\displaystyle A_{1}A_{6}=$	$\displaystyle A_{6}A_{1}=HA_{1}+IA_{2}+EA_{3}+CA_{4}+IA_{6}+GA_{6},$
	$\displaystyle A_{2}A_{3}=$	$\displaystyle A_{3}A_{2}=IA_{1}+GA_{2}+HA_{3}+IA_{4}+EA_{5}+CA_{6},$
	$\displaystyle A_{2}A_{4}=$	$\displaystyle A_{4}A_{2}=BA_{1}+HA_{2}+JA_{3}+GA_{4}+FA_{5}+IA_{6},$
	$\displaystyle A_{2}A_{5}=$	$\displaystyle A_{5}A_{1}2=(2l+1)A_{0}+HA_{1}+AA_{2}+GA_{3}+HA_{4}+AA_{5}+GA_{6},$
	$\displaystyle A_{2}A_{6}=$	$\displaystyle A_{6}A_{2}=JA_{1}+GA_{2}+FA_{3}+IA_{4}+BA_{5}+HA_{6},$
	$\displaystyle A_{3}A_{4}=$	$\displaystyle A_{4}A_{3}=CA_{1}+IA_{2}+GA_{3}+HA_{4}+IA_{5}+EA_{6},$
	$\displaystyle A_{3}A_{5}=$	$\displaystyle A_{5}A_{3}=IA_{1}+BA_{2}+HA_{3}+JA_{4}+GA_{5}+FA_{6},$
	$\displaystyle A_{3}A_{6}=$	$\displaystyle A_{6}A_{3}=(2l+1)A_{0}+GA_{1}+HA_{2}+AA_{3}+GA_{4}+HA_{5}+AA_{6},$
	$\displaystyle A_{4}A_{5}=$	$\displaystyle A_{5}A_{4}=EA_{1}+CA_{2}+IA_{3}+GA_{4}+HA_{5}+IA_{6},$
	$\displaystyle A_{4}A_{6}=$	$\displaystyle A_{6}A_{4}=FA_{1}+IA_{2}+BA_{3}+HA_{4}+JA_{5}+GA_{6},$
	$\displaystyle A_{5}A_{6}=$	$\displaystyle A_{6}A_{5}=IA_{1}+EA_{2}+CA_{3}+IA_{4}+GA_{5}+HA_{6},$

where $A_{i}^{T}$ means the transpose of the matrix $A_{i}$ , $i=0,1,\cdots,6.$

$\mathbf{Proof.}$ The proof can be drawn directly from the definitions of $A_{i}$ and Lemma 3. $\blacksquare$

Theorem 1

Suppose $p=12l+7=6f+1$ is an odd prime. Then

\displaystyle\sum_{i=0}^{6}m_{i}A_{i}(\sum_{i=0}^{6}m_{i}A_{i})^{T}=\sum_{i=0}^{6}D_{i}(\vec{m})A_{i},

where

	$\displaystyle D_{0}(\vec{m})=$	$\displaystyle m_{0}^{2}+f(m_{1}^{2}+m_{2}^{2}+m_{3}^{2}+m_{4}^{2}+m_{5}^{2}+m_{6}^{2}),$
	$\displaystyle D_{1}(\vec{m})=D_{4}(\vec{m})=$	$\displaystyle(m_{0}m_{4}+m_{0}m_{1})+(m_{1}^{2}+m_{1}m_{4}+m_{4}^{2})A+(m_{1}m_{2}+m_{4}m_{5}+m_{3}m_{6})B$
		$\displaystyle+(m_{1}m_{3}+m_{4}m_{6}+m_{2}m_{5})C+m_{1}m_{4}D+(m_{3}m_{6}+m_{2}m_{4}+m_{1}m_{5})E$
		$\displaystyle+(m_{2}m_{5}+m_{3}m_{4}+m_{1}m_{6})F+(m_{1}m_{2}+m_{1}m_{5}+m_{2}m_{4}+m_{3}^{2}+m_{4}m_{5}+m_{6}^{2})G$
		$\displaystyle+(m_{1}m_{3}+m_{1}m_{6}+m_{2}^{2}+m_{3}m_{4}+m_{5}^{2}+m_{4}m_{6})H+(m_{2}m_{6}+m_{2}m_{3}$
		$\displaystyle+m_{3}m_{5}+m_{3}m_{5}+m_{5}m_{6}+m_{2}m_{6})I+(m_{2}m_{3}+m_{5}m_{6})J,$
	$\displaystyle D_{2}(\vec{m})=D_{5}(\vec{m})=$	$\displaystyle m_{0}m_{5}+m_{0}m_{2}+(m_{2}^{2}+m_{5}^{2}+m_{2}m_{5})A+(m_{1}m_{4}+m_{2}m_{3}+m_{5}m_{6})B$
		$\displaystyle+(m_{2}m_{4}+m_{1}m_{5}+m_{3}m_{6})C+m_{2}m_{5}D+(m_{1}m_{4}+m_{3}m_{5}+m_{2}m_{6})E$
		$\displaystyle+(m_{1}m_{2}+m_{3}m_{6}+m_{4}m_{5})F+(m_{1}^{2}+m_{2}m_{3}+m_{2}m_{6}+m_{4}^{2}+m_{3}m_{5}+m_{5}m_{6})G$
		$\displaystyle+(m_{1}m_{5}+m_{1}m_{2}+m_{3}^{2}+m_{2}m_{4}+m_{6}^{2}+m_{4}m_{5})H$
		$\displaystyle+(m_{1}m_{3}+m_{1}m_{3}+m_{3}m_{4}+m_{4}m_{6}+m_{1}m_{6}+m_{4}m_{6})I+(m_{1}m_{6}+m_{3}m_{4})J,$
	$\displaystyle D_{3}(\vec{m})=D_{6}(\vec{m})=$	$\displaystyle m_{0}m_{6}+m_{0}m_{3}+(m_{3}^{2}+m_{3}m_{6}+m_{6}^{2})A+(m_{2}m_{5}+m_{3}m_{4}+m_{1}m_{6})B$
		$\displaystyle+(m_{1}m_{4}+m_{3}m_{5}+m_{2}m_{6})C+m_{3}m_{6}D+(m_{1}m_{3}+m_{2}m_{5}+m_{4}m_{6})E$
		$\displaystyle+(m_{2}m_{3}+m_{1}m_{4}+m_{5}m_{6})F+(m_{1}m_{6}+m_{2}^{2}+m_{1}m_{3}+m_{3}m_{4}+m_{4}m_{6}+m_{5}^{2})G$
		$\displaystyle+(m_{1}^{2}+m_{2}m_{6}+m_{2}m_{3}+m_{3}m_{5}+m_{4}^{2}+m_{5}m_{6})H$
		$\displaystyle+(m_{1}m_{2}+m_{1}m_{5}+m_{2}m_{4}+m_{2}m_{4}+m_{1}m_{5}+m_{4}m_{5})I+(m_{1}m_{2}+m_{4}m_{5})J.$

$\mathbf{Proof.}$ The result can be obtained directly from Lemma 5. $\blacksquare$

3 Sextic residue double circulant self-dual codes

3.1 General sextic residue double circulant self-dual codes

Theorem 2

Suppose $p=12l+7$ is an odd prime. Let $\alpha\in\mathrm{GF}(q)$ , and $\vec{m}\in\mathrm{GF}(q)^{7}$ . Then

(a) the code with the generator matrix $P_{p}(\vec{m})$ is a self-dual code on $\mathrm{GF}(q)$ if and only if the following holds

D_{0}(\vec{m})=-1,D_{1}(\vec{m})=D_{2}(\vec{m})=D_{3}(\vec{m})=0.

(b) the code with the generator matrix $B_{p}(\alpha,\vec{m})$ is a self-dual code on $\mathrm{GF}(q)$ if and only if the following holds

	$\displaystyle\alpha^{2}+p=-1,$	$\displaystyle-\alpha+m_{0}+f(m_{1}+m_{2}+m_{3}+m_{4}+m_{5}+m_{6})=0,$
	$\displaystyle D_{0}(\vec{m})=-2,$	$\displaystyle D_{1}(\vec{m})=D_{2}(\vec{m})=D_{3}(\vec{m})=-1.$

$\mathbf{Proof.}$ According Theorem 1, we have

\displaystyle P_{p}(\vec{m})P_{p}(\vec{m})^{T}=

\displaystyle A_{0}+\sum_{i=0}^{6}D_{i}(\vec{m})A_{i}.

The following can be obtained according to the definitions of self-dual codes.

B_{p}(\alpha,\vec{m})B_{p}(\alpha,\vec{m})^{T}=\begin{pmatrix}I_{p+1}&K\end{pmatrix}\begin{pmatrix}I_{p+1}\\ K^{T}\end{pmatrix}=I_{p+1}+KK^{T},

where

	$\displaystyle KK^{T}$	$\displaystyle=\left(\begin{array}[]{cccccc}\alpha&1&\cdots&1\\ -1&&&\\ \vdots&&R&\\ -1&&&\end{array}\right)\left(\begin{array}[]{cccccc}\alpha&-1&\cdots&-1\\ 1&&&\\ \vdots&&R^{T}&\\ 1&&&\end{array}\right)$
		$\displaystyle=\left(\begin{array}[]{cccccc}\alpha^{2}+p&S&\cdots&S\\ S&&&\\ \vdots&&X&\\ S&&&\end{array}\right),$

\displaystyle X=J_{p}+\sum_{i=0}^{6}D_{i}(\vec{m})A_{i},S=-\alpha+m_{0}+\frac{p-1}{6}(m_{1}+m_{2}+m_{3}+m_{4}+m_{5}+m_{6}).

The result can be obtained from the definition of self-dual codes. $\blacksquare$

We now construct an infinite family of sextic residue double circulant self-dual codes on $\mathrm{GF}(q)$ . In preparation, we have the following theorem.

Theorem 3

Let p is an odd prime and $p=24k+19$ , $k$ is a nonnegative integer. Suppose $g$ is the primitive root of $p$ ,and satisfies $g^{m}\equiv 2\bmod p$ . If $p=x^{2}+3y^{2}$ , $x\equiv 1\bmod 3$ , $y\equiv-m\bmod p$ . Especially, we can reach one of the following equations

		$\displaystyle(a)\qquad A\equiv B\equiv C\equiv E\equiv F\equiv 0\bmod 2,D\equiv 1\bmod 2,I\equiv J\equiv 1\bmod 2,G+H\equiv 1\bmod 2,$
		$\displaystyle(b)\qquad A\equiv C\equiv D\equiv E\equiv F\equiv J\equiv 0\bmod 2,B\equiv I\equiv 1\bmod 2,G+H\equiv 1\bmod 2,$
		$\displaystyle(c)\qquad A\equiv B\equiv C\equiv D\equiv E\equiv J\equiv 0\bmod 2,F\equiv I\equiv 1\bmod 2,G+H\equiv 1\bmod 2.$

$\mathbf{Proof.}$ According Theorem 4, when $y=1\bmod 3$ , let $x=3a+1$ , $y=3b+1$ . Then

p=12l+7=x^{2}+3y^{2}=9a^{2}+1+6a+3(b^{2}+1+6b)

(3)

From Lemma 4, we know $B\equiv E$ . Then

3(B+E)=\frac{48k+38+2-4(3a+1)-24(3b+1)}{12}=4k+1-a-6b\equiv 0\bmod 2.

(4)

From Equation (4), we have $a+1\equiv 0\bmod 2$ . And from Equation (3), we have $a^{2}+b^{2}\equiv 1\bmod 2$ . Thus $a$ is odd, while $b$ is even. Let $a=2c+1$ , $b=2d$ . So $x=6c+4$ , $y=6d+1$ . Then

12l+7=(6c+4)^{2}+3(6d+1)^{2}.

Namely, $l\equiv(c+1)\bmod 2$ .

Bring it into the cyclotomic numbers

\left\{\begin{aligned} &3A=\frac{12l+7-11-2(6c+4)}{12}=l-c-1\equiv 0\bmod 2,\\ &3B=\frac{12l+7+1-2(6c+4)-12(6d+1)}{12}=l-1-c-6d\equiv 0\bmod 2,\\ &3C=\frac{12l+7+1-8(6c+4)+12(6d+1)}{12}=l-1-4c+6d\equiv 0\bmod 2,\\ &3D=\frac{12l+7+1+10(6c+4)+12(6d+1)}{12}=l+5+5c+6d\equiv 0\bmod 2,\\ &3F=\frac{12l+7+1+4(6c+4)}{12}=l+2c+2\equiv 1\bmod 2.\\ \end{aligned}\right.

From Lemma 4, we know $I=F$ , $J=C$ , $B=E$ .

When $y=2\bmod 3$ , let $x=3a+1$ , $y=3b+2$ , we have

p=12l+7=x^{2}+3y^{2}=9a^{2}+1+6a+3(b^{2}+1+6b).

(5)

From Lemma 4, we know $C\equiv F$ . Then

3(C+F)=\frac{48k+84-12a+72b}{12}=4k+7-a+6b\equiv 0\bmod 2.

(6)

From Equation (6), we have $a+1\equiv 0\bmod 2$ . And from Equation (5), we have $a^{2}+b^{2}\equiv 0\bmod 2$ . Thus $a$ , $b$ are odd. Let $a=2c+1$ , $b=2d+1$ . So $x=6c+4$ , $y=6d+5$ . Then

12l+7=9(2c+1)^{2}+1+6(2c+1)+3[9(2d+1)^{2}+1+6(2d+1)].

Namely, $l\equiv(c+1)\bmod 2$ .

Bring it into the cyclotomic numbers

\left\{\begin{aligned} &3A=\frac{12l+7-11-2(6c+4)}{12}=l-c-1\equiv 0\bmod 2,\\ &3B=\frac{12l+7+1+4(6c+4)}{12}=l+2c+2\equiv 1\bmod 2,\\ &3C=\frac{12l+7+1-2(6c+4)+12(6d+5)}{12}=l+5-c+6d\equiv 0\bmod 2,\\ &3D=\frac{12l+7+1+10(6c+4)-12(6d+5)}{12}=l-1+5c-6d\equiv 0\bmod 2,\\ &3E=\frac{12l+7+1-8(6c+4)-12(6d+5)}{12}=l-7-4c-6d\equiv 0\bmod 2,\\ &3J=\frac{12l+7+1-8(6c+4)+12(6d+5)}{12}=l-4c+6d+3\equiv 0\bmod 2.\\ \end{aligned}\right.

From Lemma 4, we know $I=B$ , $F=C$ , $G+H\equiv 1\bmod 2$ .
When $y=0\bmod 3$ , let $x=3a+1$ , $y=3b$ . We have

p=12l+7=x^{2}+3y^{2}=9a^{2}+1+6a+9b^{2}.

(7)

From Lemma 4, we know $I\equiv J$ . Then

3(I+J)=\frac{48k-12a+36b}{12}=4k-a+3\equiv 0\bmod 2.

(8)

From Equation (8), we have $a+1\equiv 0\bmod 2$ . And from Equation (7), $a^{2}+b^{2}\equiv 0\bmod 2$ . Thus $a$ , $b$ are odd. Let $a=2c+1$ , $b=2d+1$ . So $x=6c+4$ , $y=6d+3$ . Then

12l+7=9(2c+1)^{2}+1+6(2c+1)+3[9(2d+1)^{2}].

Namely, $l\equiv(c+1)\bmod 2$ .

Bring it into the cyclotomic numbers

\left\{\begin{aligned} &3A=\frac{12l+7-11-8(6c+4)}{12}=l-4c-3\equiv 0\bmod 2,\\ &3B=\frac{12l+7+1-2(6c+4)+12(6d+3)}{12}=l+3-c+6d\equiv 0\bmod 2,\\ &3D=\frac{12l+7+1+16(6c+4)}{12}=l+6+8c\equiv 1\bmod 2,\\ &3E=\frac{12l+7+1-2(6c+4)-12(6d+5)}{12}=l-5-c-6d\equiv 0\bmod 2,\\ &3I=\frac{12l+7+1-2(6c+4)}{12}=l-c\equiv 1\bmod 2.\end{aligned}\right.

From Lemma 4, we know $B=C$ , $E=F$ , $I=J\equiv 1\bmod 2$ , and $G+H\equiv 1\bmod 2$ . $\blacksquare$

3.2 Sextic residue double circulant self-dual codes on $\mathrm{GF}(2)$

Theorem 4

Suppose $p$ is an odd prime and $p=24k+19$ . Then we can get a sextic residue pure double circulant self-dual code on $\mathrm{GF(2)}$ with the generation matrix $P_{p}(\vec{m})$ . And in certain conditions, it can be optimal.

$\mathbf{Proof.}$ If $p=24k+19$ , from Theorem 3, let the first parameters as an example. We have

\left\{\begin{aligned} D_{0}(0,0,0,0,1,1,1)&\equiv 1\bmod 2,\\ D_{1}(0,0,0,0,1,1,1)&=A+B+C+2G+2H+I+J\equiv 0\bmod 2,\\ D_{2}(0,0,0,0,1,1,1)&=A+B+F+2G+2H+2I+2J\equiv 0\bmod 2,\\ D_{3}(0,0,0,0,1,1,1)&=A+E+F+2G+2H+I+J\equiv 0\bmod 2.\\ \end{aligned}\right.

From Theorem 2, we know the generation matrix $P_{p}(\vec{m})$ on $\mathrm{GF(2)}$ is a self-dual code with length of $2p$ . And the experimental results shows that the minimum Hamming distance is 8. So the codes whose parameters in Table 5 are optimal codes according the list in [7]. $\blacksquare$

Theorem 5

Suppose $p$ is an odd prime and $p=24k+19$ . Then we can get a a sextic residue double circulant self-dual code with boundary on $\mathrm{GF(2)}$ which length is $2p+2$ with generation matrix $B_{p}(\alpha,m_{0},m_{1},m_{2},m_{3},m_{4},m_{5},m_{6})$ . And in certain conditions, it can be optimal.
Table 1: Parameters of the code with $P_{41}$ $m_{0}$ $m_{1}$ $m_{2}$ $m_{3}$ $m_{4}$ $m_{5}$ $m_{6}$ $MHD$ 0 0 0 0 1 1 1 8 0 0 1 0 0 1 1 8 0 0 1 0 1 0 1 8 0 0 1 0 1 1 0 8 0 0 1 1 0 1 0 8 0 1 1 0 0 1 0 8 0 1 1 1 0 0 0 8 0 0 1 0 1 0 1 8 0 0 1 1 1 0 0 8 1 0 0 0 1 0 1 8 0 1 0 0 0 1 1 8 0 1 0 1 0 1 0 8 0 1 1 0 0 0 1 8 0 1 1 1 0 0 0 8 1 0 1 1 1 1 0 8 1 1 0 1 0 0 0 8 0 0 1 0 1 0 1 8 Table 2: Parameters of the code with $B_{41}$ $m_{0}$ $m_{1}$ $m_{2}$ $m_{3}$ $m_{4}$ $m_{5}$ $m_{6}$ $MHD$ 0 0 0 1 1 1 1 8 0 0 1 0 1 1 1 8 0 0 1 1 1 0 1 8 0 1 0 0 1 1 1 8 0 1 0 1 0 1 1 8 0 1 0 1 1 1 0 8 0 1 1 0 1 0 1 8 0 1 1 1 0 0 1 8 0 1 1 1 0 1 0 8 0 1 1 1 1 0 0 8 1 0 0 0 1 1 1 8 1 0 1 0 1 0 1 8 1 1 0 1 0 1 0 8 1 1 1 1 0 0 0 8

$\mathbf{Proof.}$ If $p=24k+19$ , from Theorem 3, let the first parameters as an example. We have

\left\{\begin{aligned} D_{0}(0,0,0,1,1,1,1)&\equiv 0\bmod 2,\\ D_{1}(0,0,0,1,1,1,1)&=A+2B+C+E+F+3G+3H+3I+J\equiv 1\bmod 2,\\ D_{2}(0,0,0,1,1,1,1)&=A+B+C+E+2F+G+H+I+J\equiv 1\bmod 2,\\ D_{3}(0,0,0,1,1,1,1)&=A+B+C+D+E+F+G+H+I+J\equiv 1\bmod 2,\\ \alpha^{2}+p=1\bmod 2,&\\ -\alpha+m_{0}+\frac{p-1}{6}(&m_{1}+m_{2}+m_{3}+m_{4}+m_{5}+m_{6})=0\bmod 2.\\ \end{aligned}\right.

From Theorem 2, we know the code with the generation matrix $B_{p}(\alpha,m_{0},m_{1},m_{2},m_{3},m_{4},m_{5},m_{6})$ on $\mathrm{GF(2)}$ is a self-dual code with length of $2p+2$ . And the experimental results show that the minimum Hamming distance is 8. So the codes whose parameters are in Table 5 are optimal codes from the list in [7]. $\blacksquare$

3.3 Sextic residue double circulant self-dual codes on $\mathrm{GF}(4)$

Let $\varepsilon$ satisfy $\varepsilon^{2}+\varepsilon+1=0$ and be a fixed primitive element of $\mathrm{GF(4)}$ . Then we have

Theorem 6

Suppose $p=24k+19$ is an odd prime. Then we can get a a sextic residue pure double circulant self-dual code on $\mathrm{GF(4)}$ with generation matrix $P_{p}(\varepsilon,\varepsilon^{2},1,\varepsilon^{2},0,0,\varepsilon)$ . And in certain conditions, it can be optimal or best than ever.

$\mathbf{Proof.}$ Suppose $p=24k+19$ is an odd prime. From Theorem 3, we have

	$\displaystyle D_{0}(\varepsilon,\varepsilon^{2},1,\varepsilon^{2},0,0,\varepsilon)=$	$\displaystyle(4k+3)(1+\varepsilon+1+\varepsilon^{2})\equiv 1\bmod 2,$
	$\displaystyle D_{1}(\varepsilon,\varepsilon^{2},1,\varepsilon^{2},0,0,\varepsilon)=$	$\displaystyle(A+B+C+3G+H+3I+J)\varepsilon$
		$\displaystyle+(2B+E+F+2G+2H+I+J+1)\equiv 0\bmod 2,$
	$\displaystyle D_{2}(\varepsilon,\varepsilon^{2},1,\varepsilon^{2},0,0,\varepsilon)=$	$\displaystyle(A+C+J)+\varepsilon^{2}B+E\varepsilon+(1+\varepsilon^{2})F$
		$\displaystyle+(2\varepsilon+\varepsilon^{2})(G+H)+(\varepsilon^{2}+\varepsilon)I+\varepsilon\equiv 0\bmod 2,$
	$\displaystyle D_{3}(\varepsilon,\varepsilon^{2},1,\varepsilon^{2},0,0,\varepsilon)=$	$\displaystyle(2A+C+E+F+G+3H+I+J+1)\varepsilon$
		$\displaystyle+(2A+B+D+F+2G+H+I+J)\equiv 0\bmod 2.$

From Theorem 2, we know the code generate by matrix $P_{p}(\varepsilon,\varepsilon^{2},1,\varepsilon^{2},0,0,\varepsilon)$ on $\mathrm{GF(4)}$ is a self-dual code with length of $2p$ . $\blacksquare$

As an example, Table 3.3 gives some $[38,19,11]_{4}$ linear codes by Theorem 6 which can reach the lower bound in Lemma 1 or possess the best than ever bound [7].
Table 3: Parameters of codes with $P_{19}$ $construction$ $Remark$ $P_{19}(\varepsilon^{2},\varepsilon^{2},\varepsilon,1,\varepsilon,0,0)$ Best than ever $P_{19}(\varepsilon,\varepsilon^{2},0,0,\varepsilon,\varepsilon^{2},1)$ Best than ever $P_{19}(\varepsilon,\varepsilon^{2},1,\varepsilon^{2},0,0,\varepsilon)$ Best than ever $P_{19}(\varepsilon,\varepsilon^{2},\varepsilon,1,\varepsilon^{2},1,1)$ Optimal code $P_{19}(\varepsilon^{2},\varepsilon^{2},1,\varepsilon,\varepsilon^{2},\varepsilon,\varepsilon^{2})$ Optimal code Table 4: Parameters of codes with $B_{19}$ $construction$ $Remark$ $B_{19}(0,0,1,1,\varepsilon,\varepsilon^{2},\varepsilon^{2},\varepsilon)$ Optimal code $B_{19}(0,\varepsilon,\varepsilon,\varepsilon^{2},\varepsilon,\varepsilon^{2},0)$ Optimal code $B_{19}(0,0,\varepsilon,\varepsilon,\varepsilon,\varepsilon^{2},\varepsilon,0)$ Optimal code $B_{19}(0,0,\varepsilon,\varepsilon^{2},\varepsilon^{2},\varepsilon,1,1)$ Optimal code $B_{19}(0,\varepsilon,\varepsilon^{2},\varepsilon,\varepsilon,\varepsilon,1,\varepsilon)$ Optimal code $B_{19}(0,0,\varepsilon^{2},\varepsilon^{2},\varepsilon,1,1,\varepsilon)$ Optimal code

Theorem 7

Suppose $p=24k+19$ is an odd prime. Then we can get a a sextic residue double circulant self-dual code with boundary on $\mathrm{GF(4)}$ with generation matrix $B_{p}(0,0,1,1,\varepsilon,\varepsilon^{2},\varepsilon^{2},\varepsilon)$ . And in certain conditions, it can be optimal.

$\mathbf{Proof.}$ Let $p=24k+19$ is an odd prime, then from Theorem 3,

\left\{\begin{aligned} D_{0}(0,0,1,1,\varepsilon,\varepsilon^{2},\varepsilon^{2},\varepsilon)&=(4k+3)(1+\varepsilon+\varepsilon^{2})\equiv 0\bmod 2,\\ D_{1}(0,0,1,1,\varepsilon,\varepsilon^{2},\varepsilon^{2},\varepsilon)&=(D+E+G+H+I+J)\varepsilon+(D+E+G+H+I+J)\equiv 1\bmod 2,\\ D_{2}(0,0,1,1,\varepsilon,\varepsilon^{2},\varepsilon^{2},\varepsilon)&=(C+D+G+H+I+J)\varepsilon+(C+D+G+H+I+J)\equiv 1\bmod 2,\\ D_{3}(0,0,1,1,\varepsilon,\varepsilon^{2},\varepsilon^{2},\varepsilon)&=(A+D+G+H+I+J)\varepsilon+(A+D+G+H+I+J)\equiv 1\bmod 2,\\ \alpha^{2}+p=1\bmod 2,\,\,\,\,\,\,\,&\\ -\alpha+m_{0}+\frac{p-1}{6}(m_{1}&+m_{2}+m_{3}+m_{4}+m_{5}+m_{6})=0\bmod 2.\\ \end{aligned}\right.

From Theorem 2, we know that the code with the generation matrix $B_{p}(\alpha,m_{0},m_{1},m_{2},m_{3},m_{4},m_{5},m_{6})$ on $\mathrm{GF(4)}$ is a self-dual code with length of $2p+2$ . $\blacksquare$

As an example, Table 3.3 gives some $[40,20,12]_{4}$ linear codes constructed by Theorem 7 which can reach the lower bound in Lemma 1.

4 Summary

In this paper, we construct sextic residue pure double circulant codes and sextic residue double circulant codes with boundary from sextic cyclotomy, and give the necessary and sufficient conditions for them to be self-dual. Numerical experiments show that these self-dual codes contain some optimal ones and best than ever ones on $\mathrm{GF}(2)$ and $\mathrm{GF}(4)$ . We believe that the construction method on residues of higher degrees and the ones on the generalized cyclotomy of rings are rich sources of self-dual codes with good parameters.

5 Acknowledgement

This work was supported by Fundamental Research Funds for the Central Universities (No. ZD2019-183-008), the Major Scientific and Technological Projects of CNPC under Grant ZD2019-18 (No. ZD2019-183-001), and Shandong Provincial Natural Science Foundation of China (ZR2019MF070).

References

[1] R. J. E. B. C. Berndt and K. S. Williams. Gauss and jacobi sums. Canadian Mathematical Society Series of Monographs and Advanced Texts, 1998.
[2] W. Bosma, J. Cannon, and C. Playoust. The magma algebra system i: The user language. J Symbolic Comput, 24(3-4):235–265, 1997.
[3] P. Gaborit. Quadratic double circulant codes over fields. Journal of Combinatorial Theory, 97(1):85–107, 2002.
[4] T. A. Gulliver and M. Harada. On extremal double circulant self-dual codes of lengths 90-96. Applicable Algebra in Engineering Communication and Computing, 2019.
[5] M. Karlin. New binary coding results by circulants. IEEE Transactions on Information Theory, 15(1):81–92, 1969.
[6] G. Nebe, E. M. Rains, and N. J. A. Sloane. Self-dual codes and invariant theory (algorithms and computation in mathematics). Math Nachrichten, 2006.
[7] P.Gaborit and A.Otmani. Tables of self-dual codes. vol.https: //www. unilim. fr/ pages perso/ philippe. gaborit/ SD/ index. html, 2020.
[8] V. Pless and N. J. A. Sloane. On the classification and enumeration of self-dual codes. Journal of Combinatorial Theory, 18(3):313–335, 1975.
[9] L. Qian, M. Shi, and P. Sol $\acute{e}$ . On self-dual and $\mathrm{LCD}$ quasi-twisted codes of index two over a special chain ring. Cryptography and Communications, Discrete Structures, Boolean Functions and Sequences, 11:717–734, 2019.
[10] M. Shi, D. Huang, L. Sok, and P. Sol $\acute{e}$ . Double circulant self-dual and $\mathrm{LCD}$ codes over $\mathrm{G}$ alois rings. Advances in Mathematics of Communications, 13(1):171–183, 2019.
[11] M. Shi, L. Qian, L. Sok, and P. Sol $\acute{e}$ . On self-dual negacirculant codes of index two and four. Designs, Codes and Cryptography, 11:2485–2494, 2018.
[12] M. Shi, H. Zhu, L. Qian, L. Sok, and P. Sol $\acute{e}$ . On self-dual and $\mathrm{LCD}$ double circulant and double negacirculant codes over $\mathrm{F}_{q}+u\mathrm{F}_{q}$ . Cryptography and Communications, Discrete Structures, Boolean Functions and Sequences, https://doi.org/10.1007/s12095-019-00363-9, 2019.
[13] T. Zhang and G. Ge. Fourth power residue double circulant self-dual codes. IEEE Transactions on Information Theory, 61(8):4243–4252, 2015.

Double Circulant Self-dual Codes on Sextic Cyclotomy

Abstract

1 Introduction

Lemma 1

2 Primaries

2.1 cyclotomy

Lemma 2

Lemma 3

Lemma 4

2.2 Sextic residue double circulant codes

Definition 1

Lemma 5

Theorem 1

3 Sextic residue double circulant self-dual codes

3.1 General sextic residue double circulant self-dual codes

Theorem 2

Theorem 3

3.2 Sextic residue double circulant self-dual codes on GF​(2)\mathrm{GF}(2)

Theorem 4

Theorem 5

3.3 Sextic residue double circulant self-dual codes on GF​(4)\mathrm{GF}(4)

Theorem 6

Theorem 7

4 Summary

5 Acknowledgement

References

3.2 Sextic residue double circulant self-dual codes on $\mathrm{GF}(2)$

3.3 Sextic residue double circulant self-dual codes on $\mathrm{GF}(4)$