Parameters of several families of binary duadic codes and their related codes

TL;DR

This paper explores various families of binary duadic codes, providing new lower bounds on their minimum distances and analyzing their dual and extended code parameters, advancing understanding of their error-correcting capabilities.

Contribution

It introduces several new families of binary duadic codes with bounds on minimum distances and investigates their dual and extended code parameters.

Findings

Lower bounds on minimum distances are close to the square-root bound.

Parameters of dual and extended codes are characterized.

Partially solves an open problem on minimum distance bounds.

Abstract

Binary duadic codes are an interesting subclass of cyclic codes since they have large dimensions and their minimum distances may have a square-root bound. In this paper, we present several families of binary duadic codes of length $2^m-1$ and develop some lower bounds on their minimum distances by using the BCH bound on cyclic codes, which partially solves one case of the open problem proposed in \cite{LLD}. It is shown that the lower bounds on their minimum distances are close to the square root bound. Moreover, the parameters of the dual and extended codes of these binary duadic codes are investigated.

Tables

Table 1. TABLE I : Parameters of 𝒞 [ 6 , 7 , S ] subscript 𝒞 6 7 𝑆 \mathcal{C}_{[6,7,S]} and 𝒞 [ 6 , 7 , S ¯ ] subscript 𝒞 6 7 ¯ 𝑆 \mathcal{C}_{[6,7,\bar{S}]} and their dual codes

$S$	${𝒞}_{[6,7,S]}$	${𝒞}_{[6,7,S]}^{⟂}$		$\overline{S}$	${𝒞}_{[6,7,\overline{S}]}$	${𝒞}_{[6,7,\overline{S}]}^{⟂}$
$\{0,2,3\}$	$[127,64,15]$	$[127,63,20]$		$\{1,4,5\}$	$[127,64,15]$	$[127,63,20]$
$\{0,3,5\}$	$[127,64,19]$	$[127,63,20]$		$\{1,2,4\}$	$[127,64,19]$	$[127,63,20]$
$\{0,4,5\}$	$[127,64,15]$	$[127,63,16]$		$\{1,2,3\}$	$[127,64,15]$	$[127,63,16]$

Table 2. TABLE II: Parameters of some known duadic codes

Code	Parameters	Dual code	Parameters	Reference
${\text{PRM}}_2(3,7)$	$[127,64,15]$	${\text{PRM}}_2{(3,7)}^{⟂}$	$[127,63,16]$	[1]
${𝒞}_{[2,7,\left\{0\right\}]}$	$[127,64,19]$	${𝒞}_{[2,7,\left\{0\right\}]}^{⟂}$	$[127,63,20]$	[12]
${𝒞}_{[2,7,\left\{1\right\}]}$	$[127,64,19]$	${𝒞}_{[2,7,\left\{1\right\}]}^{⟂}$	$[127,63,20]$	[12]
${𝒞}_{[4,7,\{0,1\}]}$	$[127,64,15]$	${𝒞}_{[4,7,\{0,1\}]}^{⟂}$	$[127,63,20]$	[10]
${𝒞}_{[4,7,\{2,3\}]}$	$[127,64,15]$	${𝒞}_{[4,7,\{2,3\}]}^{⟂}$	$[127,63,20]$	[10]

Equations

$${\mathcal{C}}^{\perp }=\{\mathbf{b}\in {\mathbb{F}}_q^n:\mathbf{b}{\mathbf{c}}^T=0\text{for all c∈C}\},$$

$$\bar{\mathcal{C}}=\{(c_0,c_1,\dots ,c_{n-1},c_n):(c_0,c_1,\dots ,c_{n-1})\in \mathcal{C}\text{ with }{c}_0+c_1+\cdots +c_{n-1}+c_n=0\}.$$

$$c_0+c_{1}x+c_2{x}^2+\cdots +c_{n-1}{x}^{n-1}\in {\mathbb{F}}_q[x]/⟨x^n-1⟩,$$

$$g_i(x)=\prod _{i\in S_i}(x-{\beta }^i)\text{mbox}and{\tilde{g}}_i(x)=(x-1)g_i(x)$$

$$C_s^{(2,n)}=\{s,s2,s{2}^2,\dots ,s{2}^{l_s-1}\}\mathrm{mod}n\subseteq {\mathbb{Z}}_n,$$

$$i=i_{m-1}{2}^{m-1}+i_{m-2}{2}^{m-2}+\cdots +i_{1}2+i_0$$

$$T_{[r,m,S]}=\{1\le i\le n-1:w_2(i)\mathrm{mod}r\in S\}.$$

$$g_{[r,m,S]}(x)=\prod _{i\in T_{[r,m,S]}}(x-{\alpha }^i).$$

$$T_{[6,m,S]}=-T_{[6,m,\bar{S}]}\text{ and }{T}_{[6,m,S]}\cup T_{[6,m,\bar{S}]}={\mathbb{Z}}_n\setminus \{0\}.$$

$$\gcd (a^m-1,a^l-1)=a^{\gcd (m,l)}-1,$$

$$\{av:1\le a\le 2^{(m-1)/2}+2\}\subseteq T_{[6,m,\{0,4,5\}]}.$$

$$\{av:1\le a\le 2^{(m-1)/2}+2\}\subseteq T_{[6,m,\{1,2,3\}]}.$$

$$av=2^{m-1}+2^{(m-1)/2}-2=2(2^{m-2}+2^{(m-3)/2}-1).$$

$$\bar{a}=\sum _{i=0}^{(m-3)/2}{a}_i{2}^i.$$

$$\bar{a}v=\sum _{i=1}^{(m-3)/2}{a}_i{2}^{i+(m-1)/2}+\sum _{i=0}^{(m-3)/2}(1-a_i)2^i+1.$$

$$w_2(\bar{a}v)=w_2(\bar{a})-1+1+\frac{m-1}{2}-w_2(\bar{a})=\frac{m-1}{2}\equiv 0(\mathrm{mod}6).$$

$$\gcd (v,n)=2^{\gcd ((m+1)/2,m)}-1=2^{\gcd ((m+1)/2,(m-1)/2)}-1=1.$$

$$w_2(av)=w_2(v)=\frac{m+1}{2}\equiv 1(\mathrm{mod}6).$$

$$\bar{a}=\sum _{i=0}^{(m-3)/2}{a}_i{2}^i.$$

$$\bar{a}v$$

$$=\sum _{i=1}^{(m-3)/2}{a}_i{2}^{i+(m+1)/2}+2^{(m-1)/2}+\sum _{i=0}^{(m-3)/2}(1-a_i)2^i+1.$$

$$w_2(\bar{a}v)=w_2(\bar{a})-1+2+\frac{m-1}{2}-w_2(\bar{a})=\frac{m+1}{2}\equiv 1(\mathrm{mod}6).$$

$$\{av:1\le a\le 2^{(m-1)/2}\}\subseteq T_{[6,m,\{0,4,5\}]}.$$

$$\{av:1\le a\le 2^{(m-1)/2}\}\subseteq T_{[6,m,\{1,2,3\}]}.$$

$$\{av:1\le a\le 2^{(m-1)/2}\}\subseteq T_{[6,m,\{0,1,5\}]}.$$

$$\{av:1\le a\le 2^{(m-1)/2}\}\subseteq T_{[6,m,\{2,3,4\}]}.$$

$$\{av:1\le a\le 2^{(m-1)/2}+2\}\subseteq T_{[6,m,\{0,1,5\}]}.$$

$$\{av:1\le a\le 2^{(m-1)/2}+2\}\subseteq T_{[6,m,\{2,3,4\}]}.$$

$$\{av:1\le a\le 2^{(m-1)/2}\}\subseteq T_{[6,m,\{0,1,2\}]}.$$

$$\{av:1\le a\le 2^{(m-1)/2}\}\subseteq T_{[6,m,\{3,4,5\}]}.$$

$$\{av:1\le a\le 2^{(m-1)/2}+2\}\subseteq T_{[6,m,\{0,1,2\}]}.$$

Full text

1 Introduction

In this paper, let $\mathbb{F}_{q}$ denote the finite field of order $q$, where $q$ is a power of a prime $p$. An $[n,k,d]$ linear code $\mathcal{C}$ over $\mathbb{F}_{q}$ is a $k$-dimensional subspace of $\mathbb{F}_{q}^{n}$ with minimum (Hamming) distance $d$. The dual code of $\mathcal{C}$, denoted by $\mathcal{C}^{\perp}$, is defined by

[TABLE]

where $\mathbf{b}\mathbf{c}^{T}$ is the standard inner product of two vectors $\mathbf{b}$ and $\mathbf{c}$ in $\mathbb{F}_{q}^{n}$. In addition, define the extended code $\bar{\mathcal{C}}$ to be the code

[TABLE]

It is easy to see that $\bar{\mathcal{C}}$ is an $[n+1,k]$ linear code.

The linear code $\mathcal{C}$ over $\mathbb{F}_{q}$ is said to be cyclic if $\mathbf{(}c_{0},c_{1},\ldots,c_{n-1})\in\mathcal{C}$ implies $\mathbf{(}c_{n-1},c_{0},\ldots,c_{n-2})\in\mathcal{C}$. By identifying each vector $\mathbf{(}c_{0},c_{1},\ldots,c_{n-1})\in\mathbb{F}_{q}^{n}$ with

[TABLE]

a code $\mathcal{C}$ of length $n$ over $\mathbb{F}_{q}$ corresponds to a subset of $\mathbb{F}_{q}[x]/\langle x^{n}-1\rangle$. Then $\mathcal{C}$ is a cyclic code if and only if the corresponding subset is an ideal of $\mathbb{F}_{q}[x]/\langle x^{n}-1\rangle$. Note that every ideal of $\mathbb{F}_{q}[x]/\langle x^{n}-1\rangle$ is principal. Then there is a monic polynomial $g(x)$ of the smallest degree such that $\mathcal{C}=\langle g(x)\rangle$ and $g(x)\mid(x^{n}-1)$. Then $g(x)$ is called the generator polynomial and $h(x)=(x^{n}-1)/g(x)$ is referred to as the check polynomial of $\mathcal{C}$. Throughout this paper, assume that $\gcd(q,n)=1$. Denote $m=\text{ord}_{n}(q)$, i.e., $m$ is the smallest positive integer such that $q^{m}\equiv 1\pmod{n}$. Let $\alpha$ be a primitive element of $\mathbb{F}_{q^{m}}$ and put $\beta=\alpha^{\frac{q^{m}-1}{n}}$. Then $\beta$ is a primitive $n$-th root of unity. The set $T=\{0\leq i\leq n-1:g(\beta^{i})=0\}$ is referred to as the defining set of $\mathcal{C}$ with respect to $\beta$. If $T$ contains $\delta-1$ consecutive integers, then we have the well-known BCH bound on the minimum distance of cyclic codes, i.e., $d\geq\delta$.

Let $S_{1}$ and $S_{2}$ be two subsets of $\mathbb{Z}_{n}=\{0,1,2,\ldots,n-1\}$ such that

• •

$S_{1}\cap S_{2}=\emptyset$ and $S_{1}\cup S_{2}=\mathbb{Z}_{n}\setminus\{0\}$, and

• •

both $S_{1}$ and $S_{2}$ are the union of some $2$-cyclotomic cosets modulo $n$.

If there is a unit $\mu\in\mathbb{Z}_{n}$ such that $S_{1}\mu=S_{2}$ and $S_{2}\mu=S_{1}$, then $(S_{1},S_{2},\mu)$ is called a splitting of $\mathbb{Z}_{n}$.

Let $(S_{1},S_{2},\mu)$ be a splitting of $\mathbb{Z}_{n}$. Define

[TABLE]

for $i\in\{1,2\}$. The pair of cyclic codes $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ of length $n$ over $\mathbb{F}_{2}$ with generator polynomials $g_{1}(x)$ and $g_{2}(x)$ are called odd-like duadic codes, and the pair of cyclic codes $\tilde{\mathcal{C}}_{1}$ and $\tilde{\mathcal{C}}_{2}$ of length $n$ over $\mathbb{F}_{2}$ with generator polynomials $\tilde{g}_{1}(x)$ and $\tilde{g}_{2}(x)$ are called even-like duadic codes.

By definition, the binary cyclic codes $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ have parameters $[n,(n+1)/2]$ and the binary cyclic codes $\tilde{\mathcal{C}}_{1}$ and $\tilde{\mathcal{C}}_{2}$ have parameters $[n,(n-1)/2]$. Binary cyclic codes with parameters $[n,(n\pm 1)/2]$ were investigated in [2, 7, 12, 13, 14]. It is observed that these binary cyclic codes have large dimensions. Generally, it is very hard to determine the minimum distance of a cyclic code with parameters $[n,(n\pm 1)/2]$. For now, the best one can do is to develop a good lower bound on the minimum distance of the code. For odd-like duadic codes, we have the following result [7, Theorem 6.5.2].

Theorem 1.1** (Square root bound).**

Let $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ be a pair of odd-like duadic codes of length $n$ over $\mathbb{F}_{2}$. Let $d_{o}$ be their (common) minimum odd weight. Then the following hold:

$d_{o}^{2}\geq n$. 2. 2.

If the splitting defining the duadic codes is given by $\mu=-1$, then $d_{o}^{2}-d_{o}+1\geq n$. 3. 3.

Suppose $d_{o}^{2}-d_{o}+1=n$, where $d_{o}>2$, and assume that the splitting defining the duadic codes is given by $\mu=-1$. Then $d_{o}$ is the minimum weight of both $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$.

As a generalisation of the quadratic residue codes, duadic codes were introduced and investigated in [8, 9, 11], where a number of properties are proved. In addition, Pless, Masley and Leon presented all binary duadic codes of length until 241 [11]. The total number of binary duadic codes of prime power lengths and their constructions were presented in [4] and [5]. For more information on duadic codes, the reader is referred to [7, Chapter 6].

Let $m\geq 2$ be a positive integer and let $n={2^{m}-1}$. For any $s\in\mathbb{Z}_{n}$, the $2$-cyclotomic coset of $s$ modulo $n$ is defined by

[TABLE]

where $l_{s}$ is the smallest positive integer such that $s\equiv s2^{l_{s}}\pmod{n}$. For an integer $i$ with $0\leq i\leq 2^{m}-1$, let

[TABLE]

be the $2$-adic expansion of $i$, where $i_{j}\in\{0,1\}$ with $0\leq j\leq m-1$. For any $i$ with $0\leq i\leq n-1$, define $w_{2}(i)=\sum\limits_{j=0}^{m-1}i_{j}$.

Now we recall a construction of binary duadic codes documented in [10]. Let $r\geq 2$ be a positive integer and let $n=2^{m}-1$ for an integer $m\geq 3$. Let $S$ be any proper subset of $\mathbb{Z}_{r}$. Define

[TABLE]

By definition, $T_{[r,m,S]}$ is the union of some $2$-cyclotomic cosets modulo $n$. Let $\alpha$ be a primitive element of $\mathbb{F}_{2^{m}}$. Let $\mathcal{C}_{[r,m,S]}$ denote the binary cyclic code of length $n$ with generator polynomial

[TABLE]

Let $r\geq 2$ be even and $|S|=r/2$, the code $\mathcal{C}_{[r,m,S]}$ could be a duadic code for certain odd $m$.

When $r=2$ and $|S|=1$, a family of duadic codes $\mathcal{C}_{[r,m,S]}$ were studied in [12]. 2. 2.

When $r=4$ and $|S|=2$, two families of duadic codes $\mathcal{C}_{[r,m,S]}$ were presented in [10], where the following open problem was also proposed.

Open problem. [10] Let $r\geq 6$ be an even integer. Find a subset $S$ of $\mathbb{Z}_{r}$ with $|S|=r/2$ such that $\mathcal{C}_{[r,m,S]}$ is a binary duadic code of length $n=2^{m}-1$ for infinitely many odd $m$. Determine the parameters of these duadic codes.

In this paper, we present binary duadic codes for all possible $S$ when $r=6$ and $|S|=3$, and develop some lower bounds on their minimum distances by using the BCH bound on cyclic codes. As will be seen, these lower bounds are close to the square root bound. This partially solves one case of the open problem. Moreover, the parameters of the dual and extended codes of these binary duadic codes are also investigated.

2 Constructions of all binary duadic codes for $r=6$

In this section, we follow the notation specified in Section 1. Let $S$ be any proper subset of $\mathbb{Z}_{6}$ with $|S|=3$ and $\bar{S}=\mathbb{Z}_{6}\setminus S$. It is clear that $T_{[6,m,S]}$ is the defining set of $\mathcal{C}_{[6,m,S]}$ with respect to the $n$-th primitive root of unity $\alpha$, where $n=2^{m}-1$. Note that $\mathbb{Z}_{6}^{*}=\{1,-1\}$. It then follows that $\mu=-1$ for a splitting of $\mathbb{Z}_{6}$. Moreover, we have $\left|\{S\subsetneq\mathbb{Z}_{6}:|S|=3\}\right|=\binom{6}{3}=20$. Then all binary duadic codes for $r=6$ can be constructed as follows.

When $m\equiv 1\pmod{6}$, let S\in\Big{\{}\{0,2,3\},\{0,2,4\},\{0,4,5\},\{0,3,5\}\Big{\}}. Then $\bar{S}=-S$ and $S\cup\bar{S}=\mathbb{Z}_{6}$. Note that $\omega_{2}(i)=m-\omega_{2}(n-i)$ for each $i$ with $1\leq i\leq n-1$. This leads to

[TABLE]

Thus $\mathcal{\mathcal{C}}_{[6,m,S]}$ and $\mathcal{\mathcal{C}}_{[6,m,\bar{S}]}$ form a pair of odd-like duadic codes. 2. 2.

When $m\equiv 3\pmod{6}$, let S\in\Big{\{}\{0,1,4\},\{0,1,5\},\{0,2,4\},\{0,2,5\}\Big{\}}. Then one can verify that $\mathcal{\mathcal{C}}_{[6,m,S]}$ and $\mathcal{\mathcal{C}}_{[6,m,\bar{S}]}$ form a pair of odd-like duadic codes. 3. 3.

When $m\equiv 5\pmod{6}$, let S\in\Big{\{}\{0,1,2\},\{0,1,3\},\{0,2,4\},\{0,3,4\}\Big{\}}. Then one can verify that $\mathcal{\mathcal{C}}_{[6,m,S]}$ and $\mathcal{\mathcal{C}}_{[6,m,\bar{S}]}$ form a pair of odd-like duadic codes.

It is remarked that the parameters of the codes $\mathcal{\mathcal{C}}_{[6,m,S]}$ and $\mathcal{\mathcal{C}}_{[6,m,\bar{S}]}$ for $S=\{0,2,4\}$ had been studied in [12]. Below we mainly investigate the parameters of the binary duadic codes $\mathcal{\mathcal{C}}_{[6,m,S]}$ and $\mathcal{\mathcal{C}}_{[6,m,\bar{S}]}$ except $S=\{0,2,4\}$ and their dual and extended codes. Lower bounds on their minimum distances are presented, and they are close to the square root bounds.

3 Some auxiliary results

In this section, we will present some necessary auxiliary results on the defining sets of binary duadic cyclic codes, which play an important role in developing lower bounds on minimum distances of the binary codes. The following well-known lemma will be employed later.

Lemma 3.1**.**

Let $l$ and $m$ be two positive integers. Then

[TABLE]

where $a\geq 2$ is a positive integer.

Below we divide into three cases according to the construction of the binary duadic codes.

A. The case: $m\equiv 1\pmod{6}$

When $S=\{0,4,5\}$, we have two subcases: $m\equiv 1\pmod{12}$ and $m\equiv 7\pmod{12}$.

Lemma 3.2**.**

Let $m\equiv 1\pmod{12}\geq 13$. Then we have the following.

If $v=2^{(m-1)/2}-1$, then $\gcd(v,n)=1$ and

[TABLE] 2. 2.

If $v=2^{(m+1)/2}-1$, then $\gcd(v,n)=1$ and

[TABLE]

Proof.

If $v=2^{(m-1)/2}-1$, it follows from Lemma 3.1 that $\gcd(v,n)=1$. When $a=2^{(m-1)/2}+2$, we have

[TABLE]

Consequently, $w_{2}(av)=(m-1)/2\equiv 0\pmod{6}$. When $a=2^{(m-1)/2}+1$, $av=2^{m-1}-1$ and $w_{2}(av)=(m-1)\equiv 0\pmod{6}$. When $a=2^{(m-1)/2}$, $w_{2}(av)=w_{2}(v)=(m-1)/2\equiv 0\pmod{6}$. Now we assume that $1\leq a\leq 2^{(m-1)/2}-1$. Let $a=2^{l}\bar{a}$, where $\bar{a}$ is odd and $l\geq 0$ is an integer. Then we have $1\leq a\leq 2^{(m-1)/2}-1$ and the $2$-adic expansion of $\bar{a}$ given by

[TABLE]

Since $\bar{a}$ is odd, $a_{0}=1$. We have

[TABLE]

It then follows that

[TABLE]

The desired conclusion on the first case then follows.

If $v=2^{(m+1)/2}-1$, it follows from Lemma 3.1 that

[TABLE]

When $a=2^{(m-1)/2}$, it is easy to see that

[TABLE]

Furthermore, one can similarly check that $w_{2}(av)\equiv 1\pmod{6}$ for $a=2^{(m-1)/2}+1$ and $2^{(m-1)/2}+2$. Next, we assume that $1\leq a\leq 2^{(m-1)/2}-1$. Let $a=2^{l}\bar{a}$, where $\bar{a}$ is odd and $l\geq 0$ is an integer. Then we have $1\leq\bar{a}\leq 2^{(m-1)/2}-1$. Let the 2-adic expansion of $\bar{a}$ be given by

[TABLE]

Since $\bar{a}$ is odd, $a_{0}=1$. Then

[TABLE]

As a result, we have

[TABLE]

This completes the proof. ∎

Lemma 3.3**.**

Let $m\equiv 7\pmod{12}\geq 7$. Then we have the following.

If $v=2^{(m+1)/2}-1$, then $\gcd(v,n)=1$ and

[TABLE] 2. 2.

If $v=2^{(m-1)/2}-1$, then $\gcd(v,n)=1$ and

[TABLE]

Proof.

The proof is very similar to that of Lemma 3.2 and omitted here. ∎

When $S=\{0,2,3\}$ and $\{0,3,5\}$, we have the following lemma on the defining sets $T_{[6,m,S]}$ and $T_{[6,m,\bar{S}]}$.

Lemma 3.4**.**

Let $m\equiv 1\pmod{6}\geq 7$. Then we have the following.

If $v=2^{(m-1)/2}-1$, then $\gcd(v,n)=1$ and

• •

$\{av:1\leq a\leq 2^{(m-1)/2}+2\}\subseteq T_{[6,m,\{0,2,3\}]}$,

• •

$\{av:1\leq a\leq 2^{(m-1)/2}+2\}\subseteq T_{[6,m,\{0,3,5\}]}$. 2. 2.

If $v=2^{(m+1)/2}-1$, then $\gcd(v,n)=1$ and

• •

$\{av:1\leq a\leq 2^{(m-1)/2}+2\}\subseteq T_{[6,m,\{1,4,5\}]}$,

• •

$\{av:1\leq a\leq 2^{(m-1)/2}+2\}\subseteq T_{[6,m,\{1,2,4\}]}$.

Proof.

The proof is very similar to that of Lemma 3.2 and omitted here. ∎

B. The case: $m\equiv 3\pmod{6}$

Lemma 3.5**.**

Let $m\equiv 3\pmod{12}\geq 3$. Then we have the following.

If $v=2^{(m-1)/2}-1$, then $\gcd(v,n)=1$ and

[TABLE] 2. 2.

If $v=2^{(m+1)/2}-1$, then $\gcd(v,n)=1$ and

[TABLE]

Proof.

The proof is very similar to that of Lemma 3.2 and omitted here. ∎

Lemma 3.6**.**

Let $m\equiv 9\pmod{12}\geq 9$. Then we have the following.

If $v=2^{(m+1)/2}-1$, then $\gcd(v,n)=1$ and

[TABLE] 2. 2.

If $v=2^{(m-1)/2}-1$, then $\gcd(v,n)=1$ and

[TABLE]

Proof.

The proof is very similar to that of Lemma 3.2 and omitted here. ∎

Lemma 3.7**.**

Let $m\equiv 3\pmod{6}\geq 9$. Then we have the following.

If $v=2^{(m+1)/2}-1$, then $\gcd(v,n)=1$ and

• •

$\{av:1\leq a\leq 2^{(m-1)/2}\}\subseteq T_{[6,m,\{0,2,5\}]}$,

• •

$\{av:1\leq a\leq 2^{(m-1)/2}\}\subseteq T_{[6,m,\{0,1,4\}]}$. 2. 2.

If $v=2^{(m-1)/2}-1$, then $\gcd(v,n)=1$ and

• •

$\{av:1\leq a\leq 2^{(m-1)/2}\}\subseteq T_{[6,m,\{1,3,4\}]}$,

• •

$\{av:1\leq a\leq 2^{(m-1)/2}\}\subseteq T_{[6,m,\{2,3,5\}]}$.

Proof.

The proof is very similar to that of Lemma 3.2 and omitted here. ∎

C. The case: $m\equiv 5\pmod{6}$

Lemma 3.8**.**

Let $m\equiv 5\pmod{12}\geq 5$. Then we have the following.

If $v=2^{(m-1)/2}-1$, then $\gcd(v,n)=1$ and

[TABLE] 2. 2.

If $v=2^{(m+1)/2}-1$, then $\gcd(v,n)=1$ and

[TABLE]

Proof.

The proof is very similar to that of Lemma 3.2 and omitted here. ∎

Lemma 3.9**.**

Let $m\equiv 11\pmod{12}\geq 11$. Then we have the following.

If $v=2^{(m+1)/2}-1$, then $\gcd(v,n)=1$ and

[TABLE] 2. 2.

If $v=2^{(m-1)/2}-1$, then $\gcd(v,n)=1$ and

[TABLE]

Proof.

The proof is very similar to that of Lemma 3.2 and omitted here. ∎

Lemma 3.10**.**

Let $m\equiv 5\pmod{6}\geq 5$. Then we have the following.

If $v=2^{(m+1)/2}-1$, then $\gcd(v,n)=1$ and

• •

$\{av:1\leq a\leq 2^{(m-1)/2}+2\}\subseteq T_{[6,m,\{0,1,3\}]}$,

• •

$\{av:1\leq a\leq 2^{(m-1)/2}\}\subseteq T_{[6,m,\{0,3,4\}]}$. 2. 2.

If $v=2^{(m-1)/2}-1$, then $\gcd(v,n)=1$ and

• •

$\{av:1\leq a\leq 2^{(m-1)/2}+2\}\subseteq T_{[6,m,\{2,4,5\}]}$,

• •

$\{av:1\leq a\leq 2^{(m-1)/2}\}\subseteq T_{[6,m,\{1,2,5\}]}$.

Proof.

The proof is very similar to that of Lemma 3.2 and omitted here. ∎

4 Parameters of the codes $\mathcal{C}_{[6,m,S]}$ and their related codes

In this section, we investigate the parameters of the codes $\mathcal{C}_{[6,m,S]}$ and their related codes. We begin to consider the case that $m\equiv 1\pmod{6}\geq 7$. When $S=\{0,4,5\}$, the parameters of $\mathcal{C}_{[6,m,S]}$ and $\mathcal{C}_{[6,m,\bar{S}]}$ are treated in the following theorem.

Theorem 4.1**.**

Let $m\equiv 1\pmod{6}\geq 7$ be an integer. Then $\mathcal{C}_{[6,m,\{0,4,5\}]}$ and $\mathcal{C}_{[6,m,\{1,2,3\}]}$ form a pair of odd-like duadic codes with parameters $[2^{m}-1,2^{m-1},d]$, where

[TABLE]

Proof.

It is known that $\mathcal{C}_{[6,m,\{0,4,5\}]}$ and $\mathcal{C}_{[6,m,\{1,2,3\}]}$ form a pair of duadic codes with length $n$ and dimension $(n+1)/2$, so they have the same minimum distance $d$.

We only prove the lower bounds on minimum distannce $d$ for $m\equiv 1\bmod 12$ as it is similar to prove the desired conclusion for $m\equiv 7\bmod 12$. Denote $v=2^{\frac{m-1}{2}}-1$. It follows from Lemma 3.2 that $\gcd(v,n)=1$. Let $\overline{v}$ be the integer satisfying $v\overline{v}\equiv 1\pmod{n}$. Write $\gamma=\alpha^{\overline{v}}$. It is deduced from Lemma 3.2 that defining set of $\mathcal{C}_{[6,m,\{0,4,5\}]}$ with respect to $\gamma$ contains the set $\left\{1,2,...,2^{(m-1)/2}+2\right\}$. The lower bound on minimum distance of $\mathcal{C}_{[6,m,\{0,4,5\}]}$ then follows from the BCH bound on the cyclic codes. This completes the proof. ∎

When $S=\{0,4,5\}$, the following theorem provides lower bounds on minimum distances of the dual codes $\mathcal{C}_{[6,m,S]}^{\perp}$ and $\mathcal{C}_{[6,m,\bar{S}]}^{\perp}$.

Theorem 4.2**.**

Let $m\equiv 1\pmod{6}\geq 7$ be an integer. Then $\mathcal{C}_{[6,m,\{0,4,5\}]}^{\perp}$ and $\mathcal{C}_{[6,m,\{1,2,3\}]}^{\perp}$ form a pair of even-like duadic codes with parameters $[2^{m}-1,2^{m-1}-1,d^{\perp}]$, where

[TABLE]

Proof.

It is easily seen that the defining sets of $\mathcal{C}_{[6,m,\{0,4,5\}]}^{\perp}$ and $\mathcal{C}_{[6,m,\{1,2,3\}]}^{\perp}$ with respect to $\alpha$ are $\{0\}\cup T_{[6,m,\{0,4,5\}]}$ and $\{0\}\cup T_{[6,m,\{1,2,3\}]}$, respectively. It then follows that $\mathcal{C}_{[6,m,\{0,4,5\}]}^{\perp}$ is the even-weight subcode of $\mathcal{C}_{[6,m,\{0,4,5\}]}$ and $\mathcal{C}_{[6,m,\{1,2,3\}]}^{\perp}$ is the even-weight subcode of $\mathcal{C}_{[6,m,\{1,2,3\}]}$. The desired conclusion then follows from Theorem 4.1. ∎

When $S=\{0,4,5\}$, the following theorem presents dimensions and lower bounds on minimum distances of the extended codes $\overline{\mathcal{C}_{[6,m,\{0,4,5\}]}}$ and $\overline{\mathcal{C}_{[6,m,\{1,2,3\}]}}$.

Theorem 4.3**.**

Let $m\equiv 1\pmod{6}\geq 7$ be an integer. Then the extended codes $\overline{\mathcal{C}_{[6,m,\{0,4,5\}]}}$ and $\overline{\mathcal{C}_{[6,m,\{1,2,3\}]}}$ of $\mathcal{C}_{[6,m,\{0,4,5\}]}$ and $\mathcal{C}_{[6,m,\{1,2,3\}]}$ are self-dual and doubly-even, and they have parameters

[TABLE]

Proof.

It is well known that the extended codes of a pair of odd-like binary duadic codes are self-dual if the splitting corresponding to the pair of odd-like binary duadic codes is given by $-1$ [7, Theorem 6.4.12]. As shown earlier, $(T_{[6,m,\{0,4,5\}]},T_{[6,m,\{1,2,3\}]},-1)$ is a splitting of $\mathbb{Z}_{2^{m}-1}$. Consequently, the extended codes $\overline{\mathcal{C}_{[6,m,\{0,4,5)]}}$ and $\overline{\mathcal{C}_{[6,m,\{1,2,3)]}}$ are self-dual. It then follows from [7, Theorem 6.5.1] that the Hamming weight of each codeword in $\overline{\mathcal{C}_{[6,m,\{0,4,5\}]}}$ and $\overline{\mathcal{C}_{[6,m,\{1,2,3\}]}}$ is divisible by $4$. The remaining conclusions follow from Theorem 4.1. ∎

When $S=\{0,3,5\}$ or $\{0,2,3\}$, we have the following theorem on parameters of the codes $\mathcal{C}_{[6,m,S]}$ and $\mathcal{C}_{[6,m,\bar{S}]}$ and their related codes.

Theorem 4.4**.**

Let $m\equiv 1\pmod{6}\geq 7$ be an integer and suppose that $S=\{0,3,5\}$ or $\{0,2,3\}$.

The odd-like duadic codes $\mathcal{C}_{[6,m,S]}$ and $\mathcal{C}_{[6,m,\bar{S}]}$ have parameters

[TABLE] 2. 2.

The dual codes $\mathcal{C}_{[6,m,S]}^{\perp}$ and $\mathcal{C}_{[6,m,\bar{S}]}^{\perp}$ form a pair of even-like duadic codes with parameters

[TABLE] 3. 3.

The extended codes $\overline{\mathcal{C}_{[6,m,S]}}$ and $\overline{\mathcal{C}_{[6,m,\bar{S}]}}$ are self-dual and doubly-even, and they have parameters

[TABLE]

Proof.

It is very similar to those of Theorems 4.1, 4.2, 4.3 and omitted here. ∎

When $m\equiv 3\pmod{6}$, we have the following theorem on parameters of the codes $\mathcal{C}_{[6,m,S]}$ and $\mathcal{C}_{[6,m,\bar{S}]}$ and their dual and extended codes. It can be similarly proved and we omit the details here.

Theorem 4.5**.**

Let $m\equiv 3\pmod{6}$ be an integer.

When $S=\{0,1,5\}$, we have the following.

The odd-like duadic codes $\mathcal{C}_{[6,m,S]}$ and $\mathcal{C}_{[6,m,\bar{S}]}$ have parameters $[2^{m}-1,2^{m-1},d]$, where

[TABLE] 2. 2.

The codes $\mathcal{C}_{[6,m,S]}^{\perp}$ and $\mathcal{C}_{[6,m,\bar{S}]}^{\perp}$ form a pair of even-like duadic codes with parameters $[2^{m}-1,2^{m-1}-1,d^{\perp}]$, where

[TABLE] 3. 3.

The extended codes $\overline{\mathcal{C}_{[6,m,S]}}$ and $\overline{\mathcal{C}_{[6,m,\bar{S}]}}$ are self-dual and doubly-even, and they have parameters

[TABLE]

When $S=\{0,1,4\}$ or $\{0,2,5\}$, we have the following.

The odd-like duadic codes $\mathcal{C}_{[6,m,S]}$ and $\mathcal{C}_{[6,m,\bar{S}]}$ have parameters

[TABLE] 2. 2.

The codes $\mathcal{C}_{[6,m,S]}^{\perp}$ and $\mathcal{C}_{[6,m,\bar{S}]}^{\perp}$ form a pair of even-like duadic codes with parameters

[TABLE] 3. 3.

The extended codes $\overline{\mathcal{C}_{[6,m,S]}}$ and $\overline{\mathcal{C}_{[6,m,\bar{S}]}}$ are self-dual and doubly-even, and they have parameters

[TABLE]

When $m\equiv 5\pmod{6}$, we have the following theorem on parameters of the codes $\mathcal{C}_{[6,m,S]}$ and $\mathcal{C}_{[6,m,\bar{S}]}$ and their dual and extended codes. It can be similarly proved and we omit the details here.

Theorem 4.6**.**

Let $m\equiv 5\pmod{6}$ be an integer.

When $S=\{0,1,2\}$, we have the following.

The odd-like duadic codes $\mathcal{C}_{[6,m,S]}$ and $\mathcal{C}_{[6,m,\bar{S}]}$ have parameters $[2^{m}-1,2^{m-1},d]$, where

[TABLE] 2. 2.

The codes $\mathcal{C}_{[6,m,S]}^{\perp}$ and $\mathcal{C}_{[6,m,\bar{S}]}^{\perp}$ form a pair of even-like duadic codes with parameters $[2^{m}-1,2^{m-1}-1,d^{\perp}]$, where

[TABLE] 3. 3.

The extended codes $\overline{\mathcal{C}_{[6,m,S]}}$ and $\overline{\mathcal{C}_{[6,m,\bar{S}]}}$ are self-dual and doubly-even, and they have parameters

[TABLE]

When $S=\{0,1,3\}$, we have the following.

The odd-like duadic codes $\mathcal{C}_{[6,m,S]}$ and $\mathcal{C}_{[6,m,\bar{S}]}$ have parameters

[TABLE] 2. 2.

The codes $\mathcal{C}_{[6,m,S]}^{\perp}$ and $\mathcal{C}_{[6,m,\bar{S}]}^{\perp}$ form a pair of even-like duadic codes with parameters

[TABLE] 3. 3.

The extended codes $\overline{\mathcal{C}_{[6,m,S]}}$ and $\overline{\mathcal{C}_{[6,m,\bar{S}]}}$ are self-dual and doubly-even, and they have parameters

[TABLE]

When $S=\{0,3,4\}$, we have the following.

The odd-like duadic codes $\mathcal{C}_{[6,m,S]}$ and $\mathcal{C}_{[6,m,\bar{S}]}$ have parameters

[TABLE] 2. 2.

The codes $\mathcal{C}_{[6,m,S]}^{\perp}$ and $\mathcal{C}_{[6,m,\bar{S}]}^{\perp}$ form a pair of even-like duadic codes with parameters

[TABLE] 3. 3.

The extended codes $\overline{\mathcal{C}_{[6,m,S]}}$ and $\overline{\mathcal{C}_{[6,m,\bar{S}]}}$ are self-dual and doubly-even, and they have parameters

[TABLE]

It can be seen that the lower bounds on minimum distances of the odd-like and even-like duadic codes developed in this paper are very close to the square root bound.

Example 1**.**

Let $m=7$ and let $\alpha$ be a generator of $\mathbb{F}_{2^{7}}^{*}$ with $\alpha^{7}+\alpha+1=0$. The parameters of $\mathcal{C}_{[6,7,S]}$ and $\mathcal{C}_{[6,7,\bar{S}]}$ and their dual codes are documented in Table I.

5 Summary and concluding remarks

In this paper, we investigated the parameters of several families of binary duadic codes $\mathcal{\mathcal{C}}_{[6,m,S]}$ and their dual and extended codes, which partially solves one case (i.e. $r=6$) of the open problem proposed in [10]. The dimensions of these codes were determined explicitly and lower bounds on their minimum distances were presented.

The only known binary duadic codes with a good minimum distance or a good lower bound on their minimum distances are the following:

• •

Binary quadratic residue codes with parameters $[n,(n+1)/2,d]$ and their even-weight subcodes, where $d^{2}\geq n$ and $n\equiv\pm 1\pmod{8}$ is a prime [7, Section 6.6].

• •

The punctured binary Reed-Muller codes of order $(m-1)/2$ which has parameters $[2^{m}-1,2^{m-1},2^{(m+1)/2}-1]$, where $m$ is odd [1].

• •

Binary duadic codes $\mathcal{C}_{[2,7,\{0\}]}$ and $\mathcal{C}_{[2,7,\{1\}]}$ constructed in [12], and binary duadic codes $\mathcal{C}_{[4,7,\{0,1\}]}$ and $\mathcal{C}_{[4,7,\{2,3\}]}$ constructed in [10] (Lower bounds on their minimum distances are close to the square root bound).

The binary duadic codes constructed in this paper are not identical with the family of binary quadratic residue codes, as $2^{m}-1$ is composite for many odd $m$. Some known binary duadic codes of length $127$ are listed in Table II, where $\text{PRM}_{2}(3,7)$ means the punctured binary Reed-Muller code of length $127$ and order $3$ [1]. From Tables I and II, one can always find at least one subset $S$ such that the parameters of $\mathcal{C}_{[6,7,S]}$ are different from those of punctured binary Reed-Muller codes and the binary duadic codes presented in [12] and [10].

There are only a few families of binary duadic codes whose minimum distances have a lower bound close to the square-root bound. It was shown that the lower bounds on minimum distances of the binary duadic codes constructed in this paper are close to the square root bound. Cyclic codes have many important applications in data storage and communication systems. However, it is not easy to construct cyclic codes with a large dimension and good minimum distance when the length $n$ is the product of some small primes [14]. It would be interesting to construct more families of binary duadic codes with good minimum distances.