Constant composition codes derived from linear codes

TL;DR

This paper introduces a new class of linear codes with known weight distributions and constructs constant composition codes as their subcodes, some of which are nearly optimal.

Contribution

It presents a novel method for deriving constant composition codes from linear codes, expanding the coding theory toolkit.

Findings

Some codes are nearly optimal in performance.

Explicit constructions of constant composition codes are provided.

Weight distributions of the proposed codes are determined.

Abstract

In this paper, we propose a class of linear codes and obtain their weight distribution. Some of these codes are almost optimal. Moreover, several classes of constant composition codes(CCCs) are constructed as subcodes of linear codes.

Tables

Table 1. Table 1 : For odd m 𝑚 m , the weight distribution of 𝒞 D ​ ( α ) subscript 𝒞 𝐷 𝛼 \mathcal{C}_{D(\alpha)}

weight	frequency
$0$	$1$
$\left(p-1\right)p^{m-2}$	$p^{m-1}-1$
$\left(p-1\right)p^{m-2}+ϵ\left(\overline{\eta }\left(-1\right)+p\overline{\eta }\left(-\alpha \right)\right)p^{\frac{m-3}{2}}$	$\frac{p-1}{2}\left(p^{m-1}+\overline{\eta }\left(-1\right)ϵp^{\frac{m-1}{2}}\right)$
$\left(p-1\right)p^{m-2}+ϵ\left(-\overline{\eta }\left(-1\right)+p\overline{\eta }\left(-\alpha \right)\right)p^{\frac{m-3}{2}}$	$\frac{p-1}{2}\left(p^{m-1}-\overline{\eta }\left(-1\right)ϵp^{\frac{m-1}{2}}\right)$

Table 2. Table 2 : For even m 𝑚 m , the weight distribution of 𝒞 D ​ ( α ) subscript 𝒞 𝐷 𝛼 \mathcal{C}_{D(\alpha)}

weight	frequency
$0$	$1$
$\left(p-1\right)p^{m-2}$	$\frac{p+1}{2}{p}^{m-1}-\tau \frac{p-1}{2}{p}^{\frac{m}{2}-1}-1$
$\left(p-1\right)p^{m-2}+2\tau p^{\frac{m}{2}-1}$	$\frac{p-1}{2}\left(p^{m-1}+\tau p^{\frac{m}{2}-1}\right)$

Equations

$${\mathcal{C}}_D=\{c(a)=(\mathrm{Tr}(a{d}_1),\mathrm{Tr}(a{d}_2),\cdots ,\mathrm{Tr}(a{d}_n)|a\in {\mathbb{F}}_q\}.$$

$$1+A_{1}X+A_2{X}^2+\cdots +A_n{X}^n.$$

$$G(\eta ,{\chi }_1)=\sum _{x\in {\mathbb{F}}_q}\eta (x){\chi }_1(x),$$

$$G(\overline{\eta },{\overline{\chi }}_1)=\sum _{x\in {\mathbb{F}}_p}\overline{\eta }(x){\overline{\chi }}_1(x).$$

$$G(\eta ,{\chi }_1)=(-1{)}^{m-1}{\sqrt{-1}}^{(\frac{p-1}{2}{)}^{2}m}\sqrt{q}$$

$$G(\overline{\eta },{\overline{\chi }}_1)={\sqrt{-1}}^{(\frac{p-1}{2}{)}^2}\sqrt{p}.$$

$$\sum _{x\in {\mathbb{F}}_q}\chi (f(x))=\chi (a_0-a_1^2/(4a_2))\eta (a_2)G(\eta ,\chi ).$$

$$N_{\alpha }=\#\{x\in {\mathbb{F}}_{p^m}|\mathrm{Tr}(x^2)=\alpha \}.$$

$$N_{\alpha}=\left\{\begin{array}[]{ll}p^{m-1},&\hbox{if $m$ is odd and $\alpha=0$;}\\ p^{m-1}-(-1)^{(\frac{p-1}{2})^{2}\frac{m}{2}}(p-1)p^{\frac{m-2}{2}},&\hbox{if $m$ is even and $\alpha=0$;}\\ p^{m-1}+\overline{\eta}(-\alpha)(-1)^{(\frac{p-1}{2})^{2}(\frac{m+1}{2})}p^{\frac{m-1}{2}},&\hbox{if $m$ is odd and $\alpha\neq 0$;}\\ p^{m-1}+(-1)^{(\frac{p-1}{2})^{2}\frac{m}{2}}p^{\frac{m-2}{2}},&\hbox{if $m$ is even and $\alpha\neq 0$.}\end{array}\right.$$

$$M\le nd/\left(nd-n^2+{\omega }_0^2+{\omega }_1^2+\cdots +{\omega }_{p-1}^2\right).$$

$$M=nd/\left(nd-n^2+{\omega }_0^2+{\omega }_1^2+\cdots +{\omega }_{p-1}^2\right),$$

$$D(\alpha )=\{d\in {\mathbb{F}}_{p^m}^{\ast }|\mathrm{Tr}(d^2)=\alpha \}.$$

$${\mathcal{C}}_{D(\alpha )}=\{c(a)=(\mathrm{Tr}(a{d}_1),\mathrm{Tr}(a{d}_2),\cdots ,\mathrm{Tr}(a{d}_{n_{\alpha }})|a\in {\mathbb{F}}_{p^m}\},$$

$$\sum _{u\in {\mathbb{F}}_p^{\ast }}{\zeta }_p^{-u\alpha }\sum _{v\in {\mathbb{F}}_p^{\ast }}\sum _{x\in {\mathbb{F}}_{p^m}}{\zeta }_p^{\mathrm{Tr}(avx+u{x}^2)}$$

$$G(\eta ,{\chi }_1)\sum _{u\in {\mathbb{F}}_p^{\ast }}\eta (u){\zeta }_p^{-u\alpha }\sum _{v\in {\mathbb{F}}_p^{\ast }}{\zeta }_p^{v^2\mathrm{Tr}(-\frac{a^2}{4u})}$$

$$N(a)=|x\in {\mathbb{F}}_{p^m}|\mathrm{Tr}(x^2)=\alpha \mathrm{and}\mathrm{Tr}(ax)=0|.$$

$$N(a)$$

$$p^{2}N(a)$$

$$wt(c(a))=n_{\alpha }-N(a)$$

$$(p-1)\left(p^{m-1}+\overline{\eta }(-1)ϵp^{\frac{m-1}{2}}\right).$$

$$(p-1)\left(p^{m-1}-\overline{\eta }(-1)ϵp^{\frac{m-1}{2}}\right).$$

$$wt(c(a))=n_{\alpha }-N(a)$$

$$S_{\gamma }=\{a\in {\mathbb{F}}_{p^m}^{\ast }|\mathrm{Tr}(a^2)=\gamma \}.$$

$${\mathcal{C}}_{D(\alpha )}^{\gamma }=\{c(a)|a\in S_{\gamma }\}.$$

$$n_{\alpha }$$

$$M$$

$${\omega }_{\beta }$$

$$d$$

$$n_{\alpha }$$

$$M$$

Full text

Key Words   Linear codes, Gauss sum, Constant composition codes

1 Introduction

Let $p$ be an odd prime and $q$ be a power of $p$. A linear $[n,k,d]$ code over the finite field $\mathbb{F}_{q}$ is a $k$-dimensional subspace of $\mathbb{F}_{q}^{n}$ with minimum Hamming distance $d$. Let $D=\{d_{1},d_{2},\cdots,d_{n}\}\subseteq\mathbb{F}_{q}^{*}$, where $n$ is a positive integer. Let ${\rm Tr}$ denote the trace function from $\mathbb{F}_{q}$ to $\mathbb{F}_{p}$. We define a linear code of length $n$ over $\mathbb{F}_{p}$ by

[TABLE]

Let $A_{i}$ denote the number of codewords with Hamming weight $i$ in a linear code $\mathcal{C}$ of length $n$. The weight enumerator of $\mathcal{C}$ is defined by

[TABLE]

The sequence $(1,A_{1},\cdots,A_{n})$ is called the weight distribution of the code $\mathcal{C}$.

The construction of linear code defined by (1.1) is generic in the sense that many classes of known codes could be produced by selecting the suitable defining set $D\subseteq\mathbb{F}_{q}$. So, the corrosponing exponential sums can be computed by some technologies of finite field. Therefore, the weight distributions of a large number of linear codes (cyclic codes) were obtained (see [5, 6, 8, 9, 11, 12, 14, 15, 17, 18, 19, 20, 21, 22], and references theirin).

Let $S=\{s_{0},\cdots,s_{q-1}\}$ be an alphabet of size $q$. An $[n,M,d,(\omega_{0},\omega_{1},\cdots,\omega_{q-1})_{q}]$ constant composition code(CCC) is a subset $C\subset S^{n}$ of size $M$, minimal distance $d$ and where the element $s_{i}$ occurs exactly $\omega_{i}$ times in each codeword in $C$.

Constant composition codes were studied already in the 1960s. Both algebraic and combinatorial constructions of CCCs have been proposed. For further information, the reader is referred to [1, 2, 3, 4, 13].

The aim of this paper is to construct CCCs from linear codes. Luo and Helleseth [13] proposed a new way to obtain CCCs from some known cyclic codes. Recently, Yu and Liu [16] construct several classes of CCCs form linear codes. Following this line, we define a class of linear codes $\mathcal{C}_{D(\alpha)}$ by the set $D(\alpha)$. When $\alpha=0$, Ding and Ding [6] have already studied this kind of linear codes. So, for $\alpha\neq 0$, we investigate the weight distribution of $\mathcal{C}_{D(\alpha)}$ (see Theorem 3.3). Furthermore, we choose a kind of set $S_{\gamma}$ and obtain several classes of CCCs (see Theorem 4.1).

2 Preliminaries

Throughout this paper, we let $q=p^{m}$, where $m$ is a positive integer. Let $\eta$ and $\overline{\eta}$ be the quadratic multiplicative character on $\mathbb{F}_{q}$ and $\mathbb{F}_{p}$, respectively. Let $\chi_{1}(\cdot)=\zeta_{p}^{{\rm{\rm Tr}}(\cdot)}$ and $\overline{\chi}_{1}=\zeta_{p}^{(\cdot)}$ be the canonical additive characters on $\mathbb{F}_{q}$ and $\mathbb{F}_{p}$, respectively. We define $\eta(0)=0=\overline{\eta}(0)$, then the quadratic Gaussian sum $G(\eta,\chi_{1})$ on $\mathbb{F}_{q}$ is defined by

[TABLE]

and the quadratic Gaussian sum $G(\overline{\eta},\overline{\chi}_{1})$ on $\mathbb{F}_{p}$ is defined by

[TABLE]

The following results are well known.

Lemma 2.1**.**

[7]** Let the notations be given as above, we have

[TABLE]

and

[TABLE]

Lemma 2.2**.**

[7]** Let $\chi$ be a nontrivial additive character of $\mathbb{F}_{q}$, and let $f(x)=a_{2}x^{2}+a_{1}x+a_{0}\in\mathbb{F}_{q}[x]$ with $a_{2}\neq 0$. Then

[TABLE]

The conclusion of the following lemma is easy to obtain.

Lemma 2.3**.**

If $m$ is odd, then $\eta(a)=\overline{\eta}(a)$ for any $a\in\mathbb{F}_{p}$. If $m$ is even, then $\eta(a)=1$ for any $a\in\mathbb{F}_{p}^{*}$.

We will need the following lemma.

Lemma 2.4**.**

[6]** With the notations given as above. For each $\alpha\in\mathbb{F}_{p}$, let

[TABLE]

Then

[TABLE]

At the end of this section, we give the LFVC bound of constant composition code.

Proposition 2.5**.**

[10]** Assume $nd-n^{2}+\omega_{0}^{2}+\omega_{1}^{2}+\cdots+\omega_{p-1}^{2}>0$. Then, an $[n,M,d,(\omega_{\beta})_{\beta\in\mathbb{F}_{p}}]$ CCC satisfies the following inequality

[TABLE]

If

[TABLE]

then we call CCC is optimal.

3 A class of linear code

In this section, for a fixed $\alpha\in\mathbb{F}_{p}$, we define set

[TABLE]

The corresponding linear code is given as

[TABLE]

where $n_{\alpha}$ is the length of $\mathcal{C}_{D(\alpha)}$. In particular, when $\alpha=0$, Ding and Ding [6] investigated the weight distribution of linear code $\mathcal{C}_{D(0)}$. Furthermore, Yu and Liu constructed a class of CCCs from code $\mathcal{C}_{D(0)}$. Here, we calculate the weight distribution of linear code $\mathcal{C}_{D(\alpha)}$ for $\alpha\neq 0$ and construct CCCs form $\mathcal{C}_{D(\alpha)}$.

Let $\epsilon=(-1)^{(\frac{p-1}{2})^{2}\frac{m+1}{2}}$ for odd $m$ and $\tau=(-1)^{(\frac{p-1}{2})^{2}\frac{m}{2}}$ for even $m$, we have the following result.

Lemma 3.1**.**

With the notations given as above. Then

[TABLE]

Proof.

By Lemma 2.2, we have

[TABLE]

If ${\rm Tr}(a^{2})=0$, by Lemma 2.3, one has

[TABLE]

If ${\rm Tr}(a^{2})\neq 0$, by Lemmas 2.2 and 2.3, then (3.3) is equal to

[TABLE]

The desired conclusions then follow from Lemma 2.1. ∎

The following lemma will be employed later.

Lemma 3.2**.**

For $a\in\mathbb{F}_{p^{m}}^{*}$ and $\alpha\in\mathbb{F}_{p}$, let

[TABLE]

Then

[TABLE]

Proof.

By definition, we have

[TABLE]

The desired conclusions then follow from Lemmas 2.4 and 3.1.

∎

Now, we give the main result in this section.

Theorem 3.3**.**

Let the notations be given as above.

• •

For odd $m$, $\mathcal{C}_{D(\alpha)}$ defined by (3.1) is an $[p^{m-1}+\overline{\eta}(-\alpha)\epsilon p^{\frac{m-1}{2}},m]$ code with weight distribution in Table $1$.

where $\epsilon=(-1)^{(\frac{p-1}{2})^{2}\frac{m+1}{2}}$.

• •

For even $m$, $\mathcal{C}_{D(\alpha)}$ defined by (3.1) is an $[p^{m-1}+\tau p^{\frac{m-2}{2}},m]$ code with weight distribution in Table $2$.

where $\tau=(-1)^{(\frac{p-1}{2})^{2}\frac{m}{2}}$.

Proof.

It is easy to obtain $n_{\alpha}$ from Lemma 2.4. If $m$ is odd, for $a\neq 0$, then

[TABLE]

Note that ${\rm Tr}(a^{2})\in\mathbb{F}_{p}^{*}$ is a square element, then, by Lemma 2.4, we have the number of $a$ is

[TABLE]

Note that ${\rm Tr}(a^{2})\in\mathbb{F}_{p}^{*}$ is a non-square element, then, by Lemma 2.4, we get the number of $a$ is

[TABLE]

If $m$ is even, then

[TABLE]

Note that when $a\in\mathbb{F}_{p^{m}}^{*}$, we have $wt(c(a))>0$ for any $m$. This implies that the dimension of linear code $\mathcal{C}_{D(\alpha)}$ is $m$.

Example 3.4**.**

Let $p=5$, $m=3$ and $\alpha$ is quare. By Magma, we have $\mathcal{C}_{D(\alpha)}$ is a $[30,3,20]$ code, whcih confirms the results in Table $1$.

Let $p=7$, $m=3$ and $\alpha$ is non-square. By Magma, we have $\mathcal{C}_{D(\alpha)}$ is a $[56,3,42]$ code, whcih agrees with the results in Table $1$.

Let $p=3$, $m=6$. By Magma, we have $\mathcal{C}_{D(\alpha)}$ is a $[234,6,144]$ code, whcih agrees with the results in Table $2$.

Let $p=3$, $m=4$. By Theorem 3.3, we have $\mathcal{C}_{D(\alpha)}$ is a $[30,4,18]$ code, whcih confirms the results in Table $2$. As we known, for $n=30$, $k=4$, the best code has parameters $[30,4,19]$. This implies that $\mathcal{C}_{D(\alpha)}$ is almost optimal.

∎

4 Constant composition codes

In this section, we will construct a class of CCCs as subcodes of linear code $\mathcal{C}_{D(\alpha)}$ defined by (3.1). For each $\gamma\in\mathbb{F}_{p}$, we let

[TABLE]

Let $\alpha\in\mathbb{F}_{p}^{*}$. Define

[TABLE]

Theorem 4.1**.**

Let $m$ be even and $\tau=(-1)^{(\frac{p-1}{2})^{2}\frac{m}{2}}$. The code $\mathcal{C}_{D(\alpha)}^{\gamma}$ is a CCC with parameters $(n_{\alpha},M,d,[\omega_{\beta}]_{\beta\in\mathbb{F}_{p}})$, where

1) in the case of $\gamma=0$:

[TABLE]

2) in the case of square $\alpha\gamma$:

[TABLE]

3) in the case of non-square $\alpha\gamma$:

[TABLE]

Proof.

Note that $\alpha\neq 0$, for any $\beta\in\mathbb{F}_{p}$, by Lemmas 2.2 and 2.3, we have

[TABLE]

If ${\rm Tr}(a^{2})=0$, i.e. $\gamma=0$, then (4.7) is equal to

[TABLE]

If ${\rm Tr}(a^{2})\neq 0$, i.e. $\gamma\neq 0$, then (4.7) is equal to

[TABLE]

Case I: if $\alpha\gamma$ is a square element, then we have

[TABLE]

Case II: if $\alpha\gamma$ is a non-square element, then we have

[TABLE]

$n_{\alpha}$ is the length of linear code $\mathcal{C}_{D(\alpha)}$. Note that $M$ is the size of $S_{\gamma}$, which can be obtained from Lemma 2.4. Denote by $d_{H}(c(a_{1}),c(a_{2}))$ the Hamming distance of $c(a_{1})$ and $(a_{2})$. When $a_{1}$ and $a_{2}$ run through $S_{\gamma}$ with $a_{1}\neq a_{2}$, then $a_{1}-a_{2}$ runs through $\mathbb{F}_{p^{m}}^{*}$. Therefore, the minimal distance of $\mathcal{C}_{D(\alpha)}^{\gamma}$ is the same as that of $\mathcal{C}_{D(\alpha)}$.

The desired conclusions then follow from Lemmas 2.1 and 3.3. ∎

Corollary 4.2**.**

Let $t\in\mathbb{F}_{p}^{*}$, then

[TABLE]

Proof.

Note that

[TABLE]

By Theorem 4.1, we finish the proof. ∎

Therefore, from Theorem 4.1 and Corollary 4.2, we have the following result.

Proposition 4.3**.**

For $\gamma\neq 0$, then

[TABLE]

Remark 4.4**.**

By Proposition 4.3 and Theorem 4.1, We can check that

[TABLE]

Therefore, the LFVC bound cannot be applied to measure the optimality of these CCCs.

Acknowledgment

The authors would like to thank the referees for their comments that improved the readability of the paper. The work of L. Yu was support by research funds of HBPU(Grant No. 17xjz04R).