Ternary Primitive LCD BCH codes

Abstract.

Absolute coset leaders were first proposed by the authors which have advantages in constructing binary LCD BCH codes. As a continue work, in this paper we focus on ternary linear codes. Firstly, we find the largest, second largest, and third largest absolute coset leaders of ternary primitive BCH codes. Secondly, we present three classes of ternary primitive BCH codes and determine their weight distributions. Finally, we obtain some LCD BCH codes and calculate some weight distributions. However, the calculation of weight distributions of two of these codes is equivalent to that of Kloosterman sums.

Key words and phrases:

LCD codes, BCH codes, absolute coset leaders, Kloosterman sums.

1991 Mathematics Subject Classification:

Primary: 11T23, 94B05; Secondary: 11L05.

This paper was supported by the National Natural Science Foundation of China (No.61772015), the Foundation of Science and Technology on Information Assurance Laboratory (No.KJ-17-010) and the Foundation of Jinling Institute of Technology (No.JIT-B-202016, No.JIT-FHXM-2020). Y. Wu was sponsored by NUPTSF (No. NY220137). (Corresponding author:Yansheng Wu.)

Xinmei Huang

Department of Mathematics, Jinling Institute of Technology

Nanjing, 211169, P. R. China

State Key Laboratory of Cryptology, P. O. Box 5159

Beijing, 100878, China

Qin Yue

Department of Mathematics, Nanjing University of Aeronautics and Astronautics

Nanjing, Jiangsu, 211100, China

Yansheng Wu

School of Computer Science, Nanjing University of Posts and Telecommunications

Nanjing 210023, P. R. China

Xiaoping Shi

Department of Mathematics, Nanjing Forestry University

Nanjing 210037, P. R. China

(Communicated by )

1. Introduction

Let $\mathbb{F}_{q}$ be a finite field with $q$ elements, where $q$ is a prime power. An $[n,k,d]$ linear code $\mathcal{C}$ over $\mathbb{F}_{q}$ is a linear subspace of $\mathbb{F}_{q}^{n}$ with dimension $k$ and minimum (Hamming) distance $d$ . Let $A_{i}$ denote the number of codewords in $\mathcal{C}$ with Hamming weight $i$ . The weight enumerator of $\mathcal{C}$ is defined by $1+A_{1}z+A_{2}z^{2}+\cdots+A_{n}z^{n}.$ The sequence $(1,A_{1},A_{2},\ldots,A_{n})$ is called the weight distribution of $\mathcal{C}$ . A code $\mathcal{C}$ is $t$ -weight if the number of nonzero $A_{i}$ in the sequence $(A_{1},A_{2},\ldots,A_{n})$ is equal to $t$ .

We define the standard Euclidean inner product of the $\mathbb{F}_{q}$ -vector space $\mathbb{F}_{q}^{n}$ as follows: for ${\bf a}=(a_{0},\ldots,a_{n-1}),{\bf c}=(c_{0},\ldots,c_{n-1})$ , $\langle{\bf a},{\bf c}\rangle={\bf a}{\bf c}^{T}=\sum_{i=0}^{n-1}a_{i}c_{i}$ . Let $\mathcal{C}$ be an $[n,k]$ linear code, its dual code is defined as follows:

\mathcal{C}^{\perp}=\{{\bf a}\in\mathbb{F}_{q}^{n}:{\bf a}{\bf c}^{T}=0\mbox{ for all }{\bf c}\in\mathcal{C}\}.

If the code $\mathcal{C}$ satisfies the condition that each codeword $(c_{0},c_{1},\ldots,c_{n-1})\in\mathcal{C}$ implies $(c_{n-1},c_{0},c_{1},\ldots,c_{n-2})\in\mathcal{C}$ , then $\mathcal{C}$ is said to be a cyclic code. A cyclic code $\mathcal{C}$ of length $n$ over $\mathbb{F}_{q}$ corresponds to an ideal of the quotient ring $\mathbb{F}_{q}[x]/\langle x^{n}-1\rangle$ . Furthermore, $\mathbb{F}_{q}[x]/\langle x^{n}-1\rangle$ is a principle ideal ring, and $\mathcal{C}$ is generated by a monic divisor $g(x)$ of ${x^{n}}-1$ . In this situation, $g(x)$ is called the generator polynomial of the code $\mathcal{C}$ and we write $\mathcal{C}=\langle g(x)\rangle$ .

Let $\mathbb{Z}_{n}=\{0,1,\ldots,n-1\}$ be the ring of integers modulo $n$ . For $s\in\mathbb{Z}_{n}$ , assume that $l_{s}$ is the smallest positive integer such that $q^{l_{s}}s\equiv s\pmod{n}$ . Then the $q$ -cyclotomic coset of $s$ modulo $n$ is defined by

C_{s}=\{s,sq,\cdots,sq^{l_{s}-1}\}_{\mod n}\subset\mathbb{Z}_{n}

and $|C_{s}|=l_{s}$ . The smallest integer in $C_{s}$ is called the coset leader of $C_{s}$ (see [11]). In the paper [8], the authors gave a new definition to investigate LCD BCH codes. Define that the smallest integer in the set $\{k,n-k:k\in C_{s}\}$ is called the absolute coset leader of $C_{s}$ .

Let $m=\operatorname{ord}_{n}(q)$ be the multiplicative order of $q$ modulo $n$ and $\gamma$ a primitive element of $\mathbb{F}_{q^{m}}$ . Then $\alpha=\gamma^{\frac{q^{m}-1}{n}}$ is of order $n$ . A cyclic code $\mathcal{C}_{(q,n,\delta,b)}=\langle g(x)\rangle$ of length $n$ over $\mathbb{F}_{q}$ is called a BCH code with the designed distance $\delta$ if its generator polynomial is of the form

g(x)=\prod\limits_{i\in Z}{(x-{\alpha^{i}}}),\ Z=C_{b+1}\cup C_{b+2}\cup\cdots\cup C_{b+\delta-1},

where $Z$ is called the defining set of $\mathcal{C}_{(q,n,\delta,b)}$ . If $n=q^{m}-1$ , we call $\mathcal{C}_{(q,n,\delta,b)}$ a primitive BCH code. If $b=0$ , $\mathcal{C}_{(q,n,\delta,b)}$ is called a narrow-sense BCH code; otherwise, it is called a non-narrow-sense BCH code. The dimension of $\mathcal{C}_{(q,n,\delta,b)}$ is $\dim(\mathcal{C}_{(q,n,\delta,b)})=n-|\bigcup_{i=b+1}^{b+\delta-1}C_{i}|$ . Thus, to determine the dimension of the code $\mathcal{C}_{(q,n,\delta,b)}$ , we only need to find out all coset leaders and cardinalities of the $q$ -cyclotomic cosets.

LCD cyclic codes named reversible codes were first studied by Massey for data storage applications [19]. An application of LCD codes against side-channel attacks was investigated by Carlet and Guilley, and several constructions of LCD codes were presented in [1]. Several constructions of LCD MDS codes were presented in [2, 4, 9, 10, 20]. Tzeng and Hartmann proved that the minimum distance of a class of LCD cyclic codes is greater than the BCH bound [21]. Several investigations of LCD BCH codes were studied in [8, 11, 17, 22, 23]. Parameters and the weight distributions of BCH codes are studied in [6, 7, 16, 18, 13, 12, 15]. LCD codes in Hermitian case were studied in [2, 10]. In [3], Carlet et al. completly determined all $q$ -ary( $q>3$ ) Euclidean LCD codes and all $q^{2}$ -ary ( $q>2$ ) Hermitian LCD codes for all parameter. Some binary and ternary LCD codes were investigated in [8, 25]. In [8], the authors proposed a new conception, named absolute coset leader, and constructed some binary LCD BCH codes. In this paper, we shall investigate the ternary case.

The remainder of the paper is organized as follows. In Section 2, some fundamental definitions and results are introduced. In Section 3, the largest, second largest, and third largest absolute coset leaders are presented for ternary primitive BCH codes. In Section 4, some BCH codes and their weight distributions are presented. Also, LCD BCH codes are constructed and their parameters are determined, some weight distributions are calculated, the determination of the others is equivalent to the computing of Kloosterman sums. In Section 5, we conclude this paper.

2. Preliminaries

A linear code $\mathcal{C}$ over $\mathbb{F}_{q}$ is called a linear code with complementary dual code (LCD for short) if $\mathcal{C}\bigcap\mathcal{C}^{\perp}=\{0\}$ , where $\mathcal{C}^{\perp}$ denotes the Euclidean dual of $\mathcal{C}$ .

Let $f(x)=x^{t}+a_{t-1}x^{t-1}+\cdots+a_{1}x_{1}+a_{0}$ be a monic polynomial over $\mathbb{F}_{q}$ with $a_{0}\neq 0$ . The reciprocal polynomial of $f(x)$ is defined by $\widehat{f}(x)=a_{0}^{-1}x^{t}f(x^{-1})$ . Then we have the following lemma that characterizes LCD cyclic codes over $\mathbb{F}_{q}$ .

Lemma 2.1.

[24] Let $\mathcal{C}$ be a cyclic code of length $n$ over $\mathbb{F}_{q}$ with generator polynomial $g(x)$ and $\gcd(n,q)=1$ . Then the following statements are equivalent.

(1)

$\mathcal{C}$ is an LCD code.
(2)

$g(x)$ is self-reciprocal, i.e., $g(x)=\widehat{g}(x)$ .
(3)

$\alpha^{-1}$ is a root of $g(x)$ for every root $\alpha$ of $g(x)$ .

Let $\mathbb{F}_{q}$ be the finite field with $q$ elements, where $q$ is a power of a prime number $p$ . The canonical additive character of $\mathbb{F}_{q}$ is defined as follows:

\chi:\mathbb{F}_{q}\rightarrow\mathbb{C}^{*},\chi(x)=\zeta_{p}^{\operatorname{Tr}_{q/p}(x)},

where $\zeta_{p}=e^{\frac{2\pi i}{p}}$ is a $p$ -th primitive root of unity and $\operatorname{Tr}_{q/p}$ denotes the trace function from $\mathbb{F}_{q}$ to $\mathbb{F}_{p}$ . The orthogonal property of additive characters which can be found in [14]

\underset{x\in\mathbb{F}_{q}}{\sum}\chi(ax)=\left\{\begin{aligned} q&~{}\text{ if }~{}a=0,\\ 0&~{}\text{ if }~{}a\in\mathbb{F}_{q}^{*}.\\ \end{aligned}\right.

Let $\psi:\mathbb{F}_{q}\rightarrow\mathbb{C}^{*}$ be a multiplicative character of $\mathbb{F}_{q}^{*}$ . The trivial multiplicative character $\psi_{0}$ is defined by $\psi_{0}(x)=1$ for all $x\in\mathbb{F}_{q}^{*}$ . For two multiplicative characters $\psi$ , $\psi^{\prime}$ of $\mathbb{F}_{q}^{*}$ , we define the multiplication by setting $\psi\psi^{\prime}(x)=\psi(x)\psi^{\prime}(x)$ for all $x\in\mathbb{F}_{q}^{*}$ . Let $\bar{\psi}$ be the conjugate character of $\psi$ defined by $\bar{\psi}(x)=\overline{\psi(x)}$ , where $\overline{\psi(x)}$ denotes the complex conjugate of $\psi(x)$ . It is easy to deduce that $\psi^{-1}=\bar{\psi}$ . It is known [14] that all the multiplicative characters form a multiplication group $\hat{\mathbb{F}_{q}^{*}}$ which is isomorphic to $\mathbb{F}_{q}^{*}$ . The orthogonal property of multiplicative characters [14] is:

\underset{x\in\mathbb{F}_{q}^{*}}{\sum}\psi(x)=\left\{\begin{aligned} q-1&~{}\text{ if }~{}\psi=\psi_{0},\\ 0&~{}\text{ otherwise}.\\ \end{aligned}\right.

The Gauss sum over $\mathbb{F}_{q}$ is defined by

G(\psi,\chi)=\sum_{x\in\mathbb{F}_{q}^{*}}\chi(x)\psi(x).

It is easy to see that $G(\psi_{0},\chi)=-1$ and $G(\bar{\psi},\chi)=\psi(-1)\overline{G(\psi,\chi)}$ . Gauss sums can be viewed as the Fourier coefficients in the Fourier expansion of the restriction of $\psi$ to $\mathbb{F}_{q}^{*}$ in terms of the multiplicative characters of $\mathbb{F}_{q}$ , i.e., for $x\in\mathbb{F}_{q}^{*}$ ,

\chi(x)=\frac{1}{q-1}\sum_{x\in\hat{\mathbb{F}_{q}^{*}}}G(\bar{\psi},\chi)\psi(x).

(1)

Using (1), we can get the following results.

Lemma 2.2.

[14] Let $\chi$ be a nontrivial additive character of $\mathbb{F}_{q}$ , $n\in\mathbb{N}$ , and $\lambda$ a multiplicative character of $\mathbb{F}_{q}$ of order $d=gcd(n,q-1)$ . Then

\sum_{x\in\mathbb{F}_{q}}\chi(ax^{n}+b)=\chi(b)\sum_{j=1}^{d-1}\bar{\lambda}^{j}(a)G(\lambda^{j},\chi)

for any $a,b\in\mathbb{F}_{q}$ with $a\neq 0$ .

In general, the explicit determination of Gauss sums is a difficult problem. For future use, we state the quadratic Gauss sums here.

Lemma 2.3.

[14] Let $\mathbb{F}_{q}$ be a finite field with $q=p^{s}$ , where $p$ is an odd prime and $s\in\mathbb{N}$ . Let $\eta$ be the quadratic character of $\mathbb{F}_{q}$ and let $\chi$ be the canonical additive character of $\mathbb{F}_{q}$ . Then

G(\eta,\chi)=\left\{\begin{array}[]{ll}(-1)^{s}q^{1/2}&\mbox{ if }p\equiv 1\pmod{4},\\ (-1)^{s-1}(\sqrt{-1})^{s}q^{1/2}&\mbox{ if }p\equiv 3\pmod{4}.\end{array}\right.

3. Absolute coset leaders of ternary BCH codes

In this section, we will find the first, second and third largest absolute coset leaders of ternary cyclic BCH codes of length $n=3^{m}-1$ over $\mathbb{F}_{3}$ , where $m=\operatorname{ord}_{n}(3)$ .

In [16, 18, 23], the authors determined the largest and second largest coset leaders of BCH codes in three cases: (1) $n=q^{m}-1$ ; (2) $n=\frac{q^{m}-1}{q-1}$ ; (3) $n=q^{l}+1$ . In [8], the authors determined the largest and second largest absolute coset leaders of binary BCH codes.

Before presenting our results, we describe some notations. The $3$ -adic expansion of an integer $i\in\mathbb{Z}_{n}$ is denoted by

i=i_{0}+i_{1}3+\cdots+i_{m-1}3^{m-1}\triangleq(i_{0},i_{1},\ldots,i_{m-1}),

where each $0\leq i_{t}\leq 2$ .

According to the definition of absolute coset leaders, we can get the following proposition.

Proposition 1.

[8] Let the absolute coset leader of $C_{\delta}$ be $\delta$ and $n=q^{m}-1$ .

(1) Then $\delta\leq\frac{n}{2}$ .

(2) If $n-\delta\notin C_{\delta}$ , then $C_{n-\delta}\neq C_{\delta}$ , $|C_{n-\delta}|=|C_{\delta}|$ , and $C_{n-\delta}$ has the same absolute coset leader $\delta$ as one in $C_{\delta}$ .

Theorem 3.1.

Let $q=3$ , $m$ a positive integer, and $n=q^{m}-1$ . Then $\delta_{1}=\frac{3^{m}-1}{2}$ is the largest absolute coset leader among all $3$ -cyclotomic cosets, $C_{\delta_{1}}=\{\delta_{1}\}$ , and $|C_{\delta_{1}}|=1$ .

Proof.

We shall verify that $\delta_{1}$ is the largest absolute coset leader among all $3$ -cyclotomic cosets $C_{s},0\leq s\leq n-1$ .

There are two $3$ -adic expansions of $n$ and $\delta_{1}$ :

	$\displaystyle n$	$\displaystyle=$	$\displaystyle(2,2,2,2,\ldots,2,2,2),$		(2)
	$\displaystyle\delta_{1}$	$\displaystyle=$	$\displaystyle(1,1,1,1,\ldots,1,1).$

Firstly, we prove that $\delta_{1}$ is the absolute coset leader of the $q$ -cyclotomic cosets $C_{\delta_{1}}$ . For $1\leq l\leq m-1$ ,

3^{l}\delta_{1}\pmod{n}\equiv(1,1,1,1,\ldots,1,1)

Hence $C_{\delta_{1}}=\{\delta_{1}\}$ only has one element, i.e. $|C_{\delta_{1}}|=1$ .

Secondly, we will show that $\delta_{1}$ is the largest absolute coset leader among all $q$ -cyclotomic cosets.

For $0\leq i\leq n-1$ , there is a $3$ -adic expansion:

i=i_{0}+i_{1}3+\cdots+i_{m-1}3^{m-1}=(i_{0},i_{1},\ldots,i_{m-1}),

where each $i_{t}\in\{0,1,2\}$ , $t=0,1,\ldots,m-1$ .

If the expansion of $i$ has $0$ . Without loss of generality, let $i=(\ldots,0,\ldots)$ . Then there is an integer $l$ , $0\leq l\leq m-1$ , such that $3^{l}i\pmod{n}\equiv(\ldots,0)\in C_{i}$ , so $3^{l}i\pmod{n}<\delta_{1}$ by (3.1). Hence the absolute coset leader in $C_{i}$ is less than $\delta_{1}$ .

If the expansion of $i$ has $2$ . Similarly, let $i=(\ldots,2,\ldots)$ , there is an integer $l$ , $0\leq l\leq m-1$ , such that $n-3^{l}i\pmod{n}<\delta_{1}$ . Hence the absolute coset leader in $C_{i}$ is less than $\delta_{1}$ .

Therefore, $\delta_{1}$ is the largest absolute coset leader among all cosets.

This completes the proof. ∎

Theorem 3.2.

Let $q=3$ , $m$ a positive integer, and $n=q^{m}-1$ .

(1) If $m\geq 3$ is an odd integer, then $\delta_{2}=\frac{3^{m-1}-1}{4}+3^{m-2}$ is the second largest absolute coset leader, $C_{\delta_{2}}\neq C_{n-\delta_{2}}$ , and $|C_{\delta_{2}}|=|C_{n-\delta_{2}}|=m$ .

(2) If $m\geq 2$ is an even integer, then $\delta_{2}=\frac{3^{m}-1}{4}$ is the second largest absolute coset leader, $C_{\delta_{2}}=\{\delta_{2},n-\delta_{2}\}$ , and $|C_{\delta_{2}}|=2$ .

Proof.

(1) If $m$ is odd and the $3$ -adic expansion of $\delta_{2}$ is as follows:

\delta_{2}=\frac{3^{m-1}-1}{4}+3^{m-2}=(\underbrace{2,0,2,0,\ldots,2,0}_{(m-3)/2~{}~{}(2,0)^{\prime}s},2,1,0),

then $\delta_{2}<\delta_{1}$ .

Firstly, we prove that $\delta_{2}$ is the absolute coset leader of the $q$ -cyclotomic cosets $C_{\delta_{2}}$ and $C_{n-\delta_{2}}$ . For $1\leq l\leq m-1$ , if $l$ is odd, then

3^{l}\delta_{2}\pmod{n}\equiv(\underbrace{2,0,\ldots,2,0,}_{(l-3)/2\ (2,0)^{\prime}s}2,1,0,\underbrace{2,0,\ldots,2,0}_{(m-l)/2~{}~{}(2,0)^{\prime}s});

if $l$ is even, then

3^{l}\delta_{1}\pmod{n}\equiv(0,\underbrace{2,0,\ldots,2,0,}_{(l-4)/2~{}~{}(2,0)^{\prime}s}2,1,0,\underbrace{2,0,\ldots,2,0,}_{(m-l-1)/2~{}~{}(2,0)^{\prime}s}2).

Hence $C_{\delta_{2}}$ has $m$ distinct elements, i.e. $|C_{\delta_{2}}|=m$ , and $\delta_{2}=\min\{k,n-k:k\in C_{\delta_{2}}\}$ , which is the absolute coset leader in $C_{\delta_{2}}$ . Similarly, we can prove that $|C_{n-\delta_{2}}|=m$ , $C_{\delta_{2}}\neq C_{n-\delta_{2}}$ , and $C_{n-\delta_{2}}$ has also the absolute coset leader $\delta_{2}$ .

Secondly, we prove that $\delta_{2}$ is the second largest absolute coset leader.

For $0\leq i\leq n-1$ , there is a $3$ -adic expansion:

i=i_{0}+i_{1}3+\ldots+i_{m-1}3^{m-1}=(i_{0},i_{1},\ldots,i_{m-1}),

which has at least two elements among $0,1,2$ . Otherwise, the expansion of $i$ has only one elements of $0,1,2$ , then $i=(0,\ldots,0)<\delta_{2}$ , $i=\delta_{1}$ , or $i=n-\delta_{1}$ .

If the expansion of $i$ has a consecutive form: $(00)$ , i.e., $i=(\ldots,0,0,\ldots)$ . Then there is an integer $l$ , $0\leq l\leq m-1$ , such that $3^{l}i\pmod{n}\equiv(\ldots,0,0)\in C_{i}$ , so $3^{l}i\pmod{n}<\delta_{2}$ . Hence the absolute coset leader of $C_{i}$ is less than $\delta_{2}$ . Similarly, we can prove it if the expansion of $i$ has consecutive $(22)$ .

If the expansion of $i$ has a form: $(110)$ , i.e., $i=(\ldots,1,1,0,\ldots)$ . Then there is an integer $l$ , $0\leq l\leq m-1$ , such that $3^{l}i\pmod{n}\equiv(\ldots,1,1,0)\in C_{i}$ , so $3^{l}i\pmod{n}<\delta_{2}$ . Hence the absolute coset leader of $C_{i}$ is less than $\delta_{2}$ . Similarly, we can prove if the expansion of $i$ has a from: $(112)$ . Then there is an integer $l$ , $0\leq l\leq m-1$ , such that $3^{l}i\pmod{n}\equiv(\ldots,1,1,2)\in C_{i}$ , so $n-3^{l}i\pmod{n}<\delta_{2}$ . Hence the absolute coset leader of $C_{i}$ is less than $\delta_{2}$ .

If the expansion of $i$ has a form: $(010)$ (or $(212)$ ), then there is an integer $l$ such that $3^{l}i\pmod{n}<\delta_{2}$ (or $n-3^{l}i\pmod{n}<\delta_{2}$ , respectively). Hence the absolute coset leader of $C_{i}$ is less than $\delta_{2}$ .

If the expansion of $i$ has not any forms: $(00)$ , $(11)$ , $(22)$ , $(010)$ , and $(212)$ . We shall prove that the absolute coset leader of $C_{i}$ is less than $\delta_{2}$ . From the above, the expansion of $i$ is equivalent to insert some 1’s into the sequence $(2,0,\ldots,2,0)$ (or $(0,2,\ldots,0,2)$ ). Since $m$ is an odd integer, the number of 1’s in the expansion of $i$ is an odd integer $k$ .

If $k=1$ , i.e., the expansion of $i$ has only one form: $(210)$ (or $(021)$ ), then there is an integer $l$ , $0\leq l\leq m-1$ , such that $3^{l}i\pmod{n}\equiv\delta_{2}$ (or $n-3^{l}i\pmod{n}\equiv\delta_{2}$ , respectively).

If $k\geq 3$ , without loss of generality, there are two adjacent $(210)^{\prime}s$ in the expansion of $i$ , i.e.,

i=(\ldots,\underbrace{2,1,0},2,0,\ldots,2,0,\underbrace{2,1,0},\ldots).

Then there is an integer $l$ , $0\leq l\leq m-1$ , such that

3^{l}i\pmod{n}\equiv(\ldots,\underbrace{2,1,0},2,0,\ldots,2,0,\underbrace{2,1,0})<\delta_{2}.

Similarly, if there are two adjacent $(012)^{\prime}s$ in the expansion of $i$ , i.e.,

i=(\ldots,\underbrace{0,1,2},0,2,\ldots,0,2,\underbrace{0,1,2},\ldots).

Then there is an integer $l$ , $0\leq l\leq m-1$ , such that

n-3^{l}i\pmod{n}\equiv(\ldots,\underbrace{2,1,0},2,0,\ldots,2,0,\underbrace{2,1,0})<\delta_{2}.

Therefore $\delta_{2}$ is the second largest absolute coset leader for $m$ is odd.

(2) If $m$ is even, and the expansion of $\delta_{2}$ is as follows:

\delta_{2}=\frac{3^{m}-1}{4}=(\underbrace{2,0,2,0,\ldots,2,0}_{m/2~{}~{}(2,0)^{\prime}s}),

then $\delta_{2}<\delta_{1}$ .

Firstly, we prove that $\delta_{2}$ is the absolute coset leader of the $q$ -cyclotomic cosets $C_{\delta_{2}}$ . For $1\leq l\leq m-1$ , if $l$ is odd, then $3^{l}\delta_{2}\pmod{n}\equiv\delta_{2}$ , if $l$ is even, then $n-3^{l}\delta_{1}\pmod{n}\equiv\delta_{2}$ . Hence $C_{\delta_{2}}=\{\delta_{2},n-\delta_{2}\}$ and $|C_{\delta_{2}}|=2$ . It is obvious that $\delta_{2}$ is the absolute coset leader in $C_{\delta_{2}}$ .

Secondly, we prove that $\delta_{2}$ is the second largest absolute coset leader.

For $1\leq i\leq n-1$ , the $3$ -adic expansion of $i$ is as follows: $i=(i_{0},i_{1},\ldots,i_{m-1})$ , which has at least two elements among $0,1,2$ .

If the expansion of $i$ has a form: $(10)$ (or $(12)$ ). Then there is an integer $l$ , $0\leq l\leq m-1$ , such that $3^{l}i\pmod{n}\equiv(\ldots,1,0)\in C_{i}$ (or $3^{l}i\pmod{n}\equiv(\ldots,1,2)\in C_{i}$ ), so $3^{l}i\pmod{n}<\delta_{2}$ (or $n-3^{l}i\pmod{n}<\delta_{2}$ , respectively). Hence, the absolute coset leader in $C_{i}$ is less than $\delta_{2}$ .

If the expansion of $i$ has a consecutive form: $(11)$ . Then the expansion of $i$ has $(110)$ or $(112)$ . From the above, the absolute coset leader in $C_{i}$ is less than $\delta_{2}$ .

If the expansion of $i$ has a consecutive form: $(00)$ (or $(22)$ ). Then there is an integer $l$ , $0\leq l\leq m-1$ , such that $3^{l}i\pmod{n}\equiv(\ldots,0,0)\in C_{i}$ (or $3^{l}i\pmod{n}\equiv(\ldots,2,2)\in C_{i}$ ), so $3^{l}i\pmod{n}<\delta_{2}$ (or $n-3^{l}i\pmod{n}<\delta_{2}$ , respectively). Hence, the absolute coset leader in $C_{i}$ is less than $\delta_{2}$ .

Therefore $\delta_{2}$ is the second largest absolute coset leader.

This completes the proof. ∎

Theorem 3.3.

Let $q=3$ , $m$ a positive integer, and $n=q^{m}-1$ .

(1) If $m\equiv 0\pmod{4}$ and $m\geq 4$ , then $\delta_{3}=\frac{3^{m}-1}{5}$ is the third largest absolute coset leader, $C_{\delta_{3}}=\{\delta_{3},2\delta_{3},n-3\delta_{3},n-2\delta_{3}\}$ , and $|C_{\delta_{3}}|=4$ .

(2) If $m\equiv 2\pmod{4}$ and $m\geq 6$ , then $\delta_{3}=\frac{3^{m-6}-1}{5}+3^{m-6}+2\cdot 3^{m-5}+2\cdot 3^{m-3}+3^{m-2}$ is the third largest absolute coset leader, $C_{\delta_{3}}\neq C_{n-\delta_{3}}$ , and $|C_{\delta_{3}}|=|C_{n-\delta_{3}}|=m$ .

Proof.

(1) If $m\equiv 0\pmod{4}$ and the $3$ -adic expansion of $\delta_{3}$ is as follows:

\delta_{3}=\frac{3^{m}-1}{5}=(1+2\cdot 3+3^{2})(1+3^{4}+\ldots+3^{\frac{m-4}{4}})=(\underbrace{1,2,1,0,\ldots,1,2,1,0}_{m/4~{}(1,2,1,0)^{\prime}s}).

Firstly, it is checked that $C_{\delta_{3}}=\{\delta_{3},2\delta_{3},n-2\delta_{3},n-3\delta_{3}\}=C_{n-\delta_{3}}$ , $|C_{\delta_{3}}|=4$ and $\delta_{3}$ is the absolute coset leader of the $q$ -cyclotomic cosets $C_{\delta_{3}}$ .

Secondly, we prove that $\delta_{3}$ is the third largest absolute coset leader.

For $1\leq i\leq n-1$ , the $3$ -adic expansion of $i$ is as follows: $i=(i_{0},i_{1},\ldots,i_{m-1})$ , which has at least two elements among $0,1,2$ .

If the expansion of $i$ has a consecutive form: $(00)$ or $(22)$ . Then there is an integer $l$ , $0\leq l\leq m-1$ , such that $3^{l}i\pmod{n}\equiv(\ldots,0,0)\in C_{i}$ (or $3^{l}i\pmod{n}\equiv(\ldots,2,2)\in C_{i}$ ), so $3^{l}i\pmod{n}<\delta_{3}$ (or $n-3^{l}i\pmod{n}<\delta_{3}$ , respectively). Hence, the absolute coset leader in $C_{i}$ is less than $\delta_{3}$ .

If the expansion of $i$ has a consecutive form: $(11)$ and the expansion of $i$ has the form: $(110)$ or $(112)$ . Then there is an integer $l$ , $0\leq l\leq m-1$ , such that $3^{l}i\pmod{n}\equiv(\ldots,1,1,0)\in C_{i}$ (or $3^{l}i\pmod{n}\equiv(\ldots,1,1,2)\in C_{i}$ ), so $3^{l}i\pmod{n}<\delta_{3}$ (or $n-3^{l}i\pmod{n}<\delta_{3}$ , respectively). Hence, the absolute coset leader in $C_{i}$ is less than $\delta_{3}$ .

If the expansion of $i$ has not any consecutive form: $(00)$ , $(11)$ , or $(22)$ , and it has a form: $(010)$ or $(212)$ . Then we can easily check that the absolute coset leader of $C_{i}$ is less than $\delta_{3}$ .

If the expansion of $i$ has not any form: $(00)$ , $(11)$ , $(22)$ , $(010)$ and $(212)$ , we will prove that the absolute coset leader of $C_{i}$ is less than $\delta_{3}$ . By the assumptions, the expansion of $i$ is equivalent to insert some 1’s into the sequence $(2,0,\ldots,2,0)$ (or $(0,2,\ldots,0,2)$ ). Since $m\equiv 0\pmod{4}$ , the number of 1’s in the expansion of $i$ is an even integer $k$ .

If $k=0$ , then $i=\delta_{2}$ (or $n-\delta_{2}$ ).

If $k=\frac{m}{2}$ , then $i=\delta_{3}$ or $3i\pmod{n}\equiv\delta_{3}$ .

If $2\leq k<\frac{m}{2}$ , then $i=(\ldots,2,0,2,1,0,\ldots)$ (or $i=(\ldots,0,2,0,1,2,\ldots)$ ). Hence there is an integer $l$ , $0\leq l\leq m-1$ , such that $3^{l}i\pmod{n}\equiv(\ldots,2,0,2,1,0)\in C_{i}$ (or $3^{l}i\pmod{n}\equiv(\ldots,0,2,0,1,2)\in C_{i}$ ), so $3^{l}i\pmod{n}<\delta_{3}$ (or $n-3^{l}i\pmod{n}<\delta_{3}$ , respectively). So the absolute coset leader of $C_{i}$ is smaller than $\delta_{3}$ .

Therefore $\delta_{3}$ is the third largest absolute coset leader when $m\equiv 0\pmod{4}$ .

(2) If $m\equiv 2\pmod{4}$ and the 3-adic expansion of $\delta_{3}$ is as follows:

	$\displaystyle\delta_{3}$	$\displaystyle=$	$\displaystyle(1+2\cdot 3+3^{2})(1+3^{4}+\ldots+3^{\frac{m-10}{4}})+3^{m-6}+2\cdot 3^{m-5}+2\cdot 3^{m-3}+3^{m-2}$
		$\displaystyle=$	$\displaystyle(\underbrace{1,2,1,0,\ldots,1,2,1,0}_{(m-6)/4~{}(1,2,1,0)^{\prime}s},1,2,0,2,1,0).$

In fact, the number of $1^{\prime}s$ in the expansion of $\delta_{3}$ is $\frac{m-2}{2}$ .

Firstly, we prove that $\delta_{3}$ is the absolute coset leader of the $q$ -cyclotomic cosets $C_{\delta_{3}}$ and $C_{n-\delta_{3}}$ . For $1\leq l\leq m-1$ , $3^{l}\delta_{3}\pmod{n}$ are all different and $\delta_{3}$ is the smallest one in $C_{\delta_{3}}$ . Hence $C_{\delta_{3}}$ has $m$ distinct elements, i.e. $|C_{\delta_{3}}|=m$ , and $\delta_{3}$ is the absolute coset leader in $C_{\delta_{3}}$ . Similarly, we can prove that $|C_{n-\delta_{3}}|=m$ , $C_{\delta_{3}}\neq C_{n-\delta_{3}}$ , and $C_{n-\delta_{3}}$ has also the absolute coset leader $\delta_{3}$ .

Secondly, we prove that $\delta_{3}$ is the third largest absolute coset leader.

For $1\leq i\leq n-1$ , there is a $3$ -adic expansion: $i=(i_{0},i_{1},\ldots,i_{m-1})$ , which has at least two elements among $0,1,2$ .

If the expansion of $i$ has a consecutive form: $(11)$ and the expansion of $i$ has a form: $(110)$ or $(112)$ . Then there is an integer $l$ , $0\leq l\leq m-1$ , such that $3^{l}i\pmod{n}\equiv(\ldots,1,1,0)\in C_{i}$ (or $3^{l}i\pmod{n}\equiv(\ldots,1,1,2)\in C_{i}$ ), so $3^{l}i\pmod{n}<\delta_{3}$ (or $n-3^{l}i\pmod{n}<\delta_{3}$ , respectively). Hence, the absolute coset leader in $C_{i}$ is less than $\delta_{3}$ .

If the expansion of $i$ has not any form: $(00)$ , $(11)$ , $(22)$ , $(010)$ and $(212)$ , we will prove that the absolute coset leader of $C_{i}$ is less than $\delta_{3}$ . Similarly, by the assumptions, the expansion of $i$ is equivalent to insert some 1’s into the sequence $(2,0,\ldots,2,0)$ (or $(0,2,\ldots,0,2)$ ). Since $m\equiv 2\pmod{4}$ , the number of 1’s in the expansion of $i$ is an even integer $k$ with $0\leq k\leq\frac{m-2}{2}$ .

If $k=0$ , then $i=\delta_{2}$ .

If $k=\frac{m-2}{2}$ and the expansion of $i$ has only one form: $(202)$ or $(020)$ , i.e. $i=(\ldots,1,2,1,0,\ldots,1,0,1,\underbrace{2,0,2},1,0,\ldots)$ or $i=(\ldots,1,2,1,0,\ldots,1,2,1,\underbrace{0,2,0},1,2\ldots)$ . Then there is an integer $l$ , $0\leq l\leq m-1$ , such that $3^{l}i\pmod{n}=\delta_{3}$ or $n-3^{l}i\pmod{n}=\delta_{3}$ .

If $k=\frac{m-2}{2}$ and the expansion of $i$ has not any form: $(202)$ or $(020)$ , i.e., $i=(\ldots,2,1,\underbrace{0,2},1,0,\ldots)$ (or $i=(\ldots,0,1,\underbrace{2,0},1,2,\ldots)$ ). Then there is a integer $l$ , $0\leq l\leq m-1$ , such that $3^{l}i=(\ldots,1,0,2,1,0)<\delta_{3}$ (or $n-3^{i}i=n-(\ldots,1,2,0,1,2)<\delta_{3}$ , respectively). Hence the absolute coset leader in $C_{i}$ is less than $\delta_{3}$ .

If $2\leq k<\frac{m-2}{2}$ . We consider the following some cases.

(I) If the expansion of $i$ has one of the following six cases:

$\displaystyle i$	$\displaystyle=$	$\displaystyle(\ldots,2,1,\underbrace{0,2}_{2},1,0,\ldots),~{}i=(\ldots,0,1,\underbrace{2,0}_{2},1,2,\ldots),$
$\displaystyle i$	$\displaystyle=$	$\displaystyle(\ldots,1,\underbrace{0,2,\ldots 0,2,0,2}_{>3},1,\ldots),~{}i=(\ldots,1,\underbrace{2,0,\ldots,2,0,2,0}_{>3},1,\ldots),$
$\displaystyle i$	$\displaystyle=$	$\displaystyle(\ldots,1,\underbrace{2,0,\ldots 0,2,0,2}_{>3},1,\ldots),~{}i=(\ldots,1,\underbrace{0,2,\ldots,0,2,0}_{>3},1,\ldots),$

i.e. there are two or more than three elements between two $1^{\prime}s$ . Then there is an integer $l$ such that $3^{l}i<\delta_{3}$ or $n-3^{l}i<\delta_{3}$ .

(II) If the expansion of $i$ has a form:

i=(\ldots,1,\underbrace{2,0,2},1,\underbrace{0,2,0},1,\ldots),

where each $1$ inserts between $(2,0,2)$ and $(0,2,0)$ . Then $m=k+3k=4k$ , which is contradictory.

(III) If the expansion of $i$ has a form:

i=(\ldots,1,\underbrace{2,0,2},1,0,1,2,1,\underbrace{0,2,0},1,\ldots),

where $0$ , $2$ , $(202)$ , and $(020)$ appear between two $1^{\prime}s$ . Let the number of $(202)$ and $(020)$ in the expansion of $i$ be $t$ , then $t$ is odd.

In fact, if $(202)$ and $(020)$ are viewed as $2$ and $0$ , respectively, i.e.

i^{\prime}=(\ldots,1,\underbrace{2},1,0,1,2,1,\underbrace{0},1,\ldots).

Then by $m\equiv 2\pmod{4}$ and $k$ even, $m=2k+2t$ and $t$ is odd.

Without loss of generality, there are two adjacent $(202)^{\prime}s$ in the expansion of $i$ , i.e.,

i=(\ldots,\underbrace{2,0,2},1,0,\ldots,0,1,\underbrace{2,0,2},1,0,\ldots).

Then there is an integer $l$ such that $3^{l}i<\delta_{3}$ .

Hence the absolute coset leader in $C_{i}$ is less than $\delta_{3}$ . Therefore $\delta_{3}$ is the third largest absolute coset leader for $m\equiv 2\pmod{4}$ .

This completes the proof. ∎

4. Parameters of some BCH codes

In this section, we will first present three classes of ternary BCH codes, determine their parameters and weight distributions. Secondly, four classes of ternary LCD BCH codes are proposed, weight distributions of two of these codes are calculated and the others convert to the calculations of the Kloosterman sums.

We always assume that $n=3^{m}-1$ , $\alpha$ is a primitive element of $\mathbb{F}_{3^{m}}$ , and $C_{i}$ is the $3$ -cyclotomic coset. We shall compute the weight distributions of BCH codes.

4.1. Three classes of BCH Codes and their weight distributions

Theorem 4.1.

Let $m$ be an odd integer, $\delta_{1}=\frac{3^{m}-1}{2}$ , $\delta_{2}=\frac{3^{m-1}-1}{4}+3^{m-2}$ , $Z=\bigcup_{-\delta_{2}<s\leq\delta_{1}}C_{s}$ , and $g(x)=\prod_{i\in Z}(x-\alpha^{i})$ . Then

\mathcal{C}_{(3,3^{m}-1,\delta_{1}+\delta_{2}+1,-\delta_{2})}=\{c(a)=(\operatorname{Tr}_{3^{m}/3}(a\alpha^{\delta_{2}i}))_{i=0}^{n-1}:a\in\mathbb{F}_{3^{m}}\}

is a one-weight $[3^{m}-1,m,2\cdot 3^{m-1}]$ BCH code.

Proof.

By Theorem 3.2, $\delta_{2}$ is the second largest abstract coset leader if $m$ is odd, the parity-check polynomial of $\mathcal{C}_{(3,3^{m}-1,\delta_{1}+\delta_{2},-\delta_{2})}$ is $h(x)=\prod_{i\in C_{n-\delta_{2}}}(x-\alpha^{i})$ , which is irreducible over $\mathbb{F}_{q}$ and $\deg(h(x))=m$ , so the dimension of the code is $m$ .

For $a\in\mathbb{F}_{3^{m}}^{*}$ , let $\omega$ be a 3-th primitive root of unit in the complex field. Since $\frac{3^{m}+1}{4}\in C_{n-\delta_{2}}$ , and $(\frac{3^{m}+1}{4},3^{m}-1)=1$ , we have

			$\displaystyle W_{H}(c(a))=n-\frac{1}{3}\sum_{y\in\mathbb{F}_{3}}\sum_{i=0}^{n-1}\omega^{y\operatorname{Tr}_{3^{m}/3}(a\alpha^{\delta_{2}i})}$
		$\displaystyle=$	$\displaystyle\frac{2n}{3}-\frac{1}{3}\sum_{y\in\mathbb{F}_{3}^{*}}\sum_{i=0}^{n-1}\omega^{y\operatorname{Tr}_{3^{m}/3}(a\alpha^{\frac{3^{m}+1}{4}i})}$
		$\displaystyle=$	$\displaystyle\frac{2n}{3}-\frac{1}{3}\sum_{y\in\mathbb{F}_{3}^{}}\sum_{x\in\mathbb{F}_{3^{m}}^{}}\omega^{y\operatorname{Tr}_{3^{m}/3}(ax)}=\frac{2n}{3}-\frac{2}{3}(\sum_{x\in\mathbb{F}_{3^{m}}}\omega^{\operatorname{Tr}_{3^{m}/3}(ax)}-1)=2\cdot 3^{m-1}.$

This completes the proof.∎

Example 1.

Let $p=3$ , $m=5$ , and $n=p^{m}-1=242$ . Then the BCH code in Theorem 4.1 has weight enumerator $1+242z^{162},$ which is confirmed by Magma.

Theorem 4.2.

Let $m\geq 6$ be an even integer with $m\equiv 2\pmod{4}$ , $\delta_{1}=\frac{3^{m}-1}{2}$ , $\delta_{3}=\frac{3^{m-6}-1}{5}+3^{m-6}+2\cdot 3^{m-5}+2\cdot 3^{m-3}+3^{m-2}$ , $Z=(\bigcup_{-\delta_{3}<s\leq\delta_{1}}C_{s})$ , and $g(x)=\prod_{i\in Z}(x-\alpha^{i})$ . Then

\mathcal{C}_{(3,3^{m}-1,\delta_{1}+\delta_{3}+1,-\delta_{3})}=\{c(a)=(\operatorname{Tr}_{3^{m}/3}(a\alpha^{\delta_{3}i}))_{i=0}^{n-1}:a\in\mathbb{F}_{3^{m}}\}

is a BCH code with parameters $[3^{m}-1,m,\frac{2}{3}\cdot(3^{m}-3^{\frac{m}{2}})]$ and the weight distribution in Table 1.

Table 1
Weight Frequency 0 1 $\frac{2}{3}\cdot(3^{m}-3^{\frac{m}{2}})$ $\frac{3^{m}-1}{2}$ $\frac{2}{3}\cdot(3^{m}+3^{\frac{m}{2}})$ $\frac{3^{m}-1}{2}$

Proof.

By Theorem 3.3, $\delta_{3}$ is the third largest abstract coset leader, the parity-check polynomial of $\mathcal{C}_{(3,3^{m}-1,\delta_{1}+\delta_{3}+1,-\delta_{3})}$ is $h(x)=\prod_{i\in C_{n-\delta_{3}}}(x-\alpha^{i})$ , so the dimension of the code is $m$ .

For $a\in\mathbb{F}_{3^{m}}^{*}$ , let $\omega$ be a 3-th primitive root of unit in the complex field. By $m\equiv 2\pmod{4}$ , $\alpha^{\frac{3^{m}-1}{3-1}}\in(\mathbb{F}_{3^{m}}^{*})^{2}$ and $\mathbb{F}_{3}^{*}\subset(\mathbb{F}_{3^{m}}^{*})^{2}$ . Since $\frac{3^{m}-19}{5}\in C_{n-\delta_{3}}$ and $\gcd(\frac{3^{m}-19}{5},3^{m}-1)=2$ , for $0\neq a\in\mathbb{F}_{3^{m}}$ ,

			$\displaystyle W_{H}(c(a))=n-\frac{1}{3}\sum_{y\in\mathbb{F}_{3}}\sum_{i=0}^{n-1}\omega^{y\operatorname{Tr}_{3^{m}/3}(a\alpha^{\delta_{3}i})}=\frac{2n}{3}-\frac{1}{3}\sum_{y\in\mathbb{F}_{3}^{*}}\sum_{i=0}^{n-1}\omega^{y\operatorname{Tr}_{3^{m}/3}(a\alpha^{\frac{3^{m}-19}{5}i})}$
		$\displaystyle=$	$\displaystyle\frac{2n}{3}-\frac{1}{3}\sum_{y\in\mathbb{F}_{3}^{}}\sum_{x\in\mathbb{F}_{3^{m}}^{}}\chi(yax^{2})=\frac{2n}{3}-\frac{2}{3}(\sum_{x\in\mathbb{F}_{3^{m}}}\chi(yax^{2})-1)$
		$\displaystyle=$	$\displaystyle\frac{2\cdot 3^{m}}{3}-\frac{2}{3}\eta(a)G(\eta)$
		$\displaystyle=$	$\displaystyle\left\{\begin{array}[]{ll}\frac{2}{3}\cdot(3^{m}-3^{\frac{m}{2}}),&\mbox{ if }a\mbox{ is a square },\\ \frac{2}{3}\cdot(3^{m}+3^{\frac{m}{2}}),&\mbox{ if }a\mbox{ is not a square },\end{array}\right.$

where $\eta$ is the multiplicative character of order $2$ over $\mathbb{F}_{3^{m}}$ . Hence the frequency of the weights is easy to obtain and this completes the proof. ∎

Example 2.

Let $p=3$ , $m=6$ , and $n=p^{m}-1=728$ . Then the BCH code in Theorem 4.2 has weight enumerator $1+364z^{468}+364z^{504},$ which is confirmed by Magma.

Theorem 4.3.

Let $m$ be an integer with $m\equiv 2\pmod{4}$ , $\delta_{1}=\frac{3^{m}-1}{2}$ , $\delta_{3}=\frac{3^{m-6}-1}{5}+3^{m-6}+2\cdot 3^{m-5}+2\cdot 3^{m-3}+3^{m-2}$ , $Z=(\bigcup_{-\delta_{3}<s<\delta_{1}}C_{s})$ , and $g(x)=\prod_{i\in Z}(x-\alpha^{i})$ . Then

\mathcal{C}_{(3,3^{m}-1,\delta_{1}+\delta_{3},-\delta_{3})}=\{c(a,b)=\left(a(-1)^{i}+\operatorname{Tr}_{3^{m}/3}(b\alpha^{\delta_{3}i})\right)_{i=0}^{n-1}:a\in\mathbb{F}_{3},b\in\mathbb{F}_{3^{m}}\}

is a ternary BCH code with parameters $[3^{m}-1,m+1,\frac{2}{3}\cdot(3^{m}-3^{\frac{m}{2}})]$ and the weight distribution in Table 2.

Table 2
Weight Frequency 0 1 $\frac{2}{3}\cdot(3^{m}-3^{\frac{m}{2}})$ $\frac{3^{m}-1}{2}$ $\frac{2}{3}\cdot(3^{m}+3^{\frac{m}{2}})$ $\frac{3^{m}-1}{2}$ $\frac{1}{3}(2\cdot 3^{m}+3^{\frac{m}{2}})-1$ $3^{m}-1$ $\frac{1}{3}(2\cdot 3^{m}-3^{\frac{m}{2}})-1$ $3^{m}-1$ $3^{m}-1$ 2

Proof.

By Theorem 3.3, $\delta_{3}$ is the third largest abstract coset leader, the parity-check polynomial of $\mathcal{C}$ is $h(x)=(x+1)\prod_{i\in C_{n-\delta_{3}}}(x-\alpha^{i})$ , so the dimension of the code is $m+1$ .

Let $\omega$ be a 3-th primitive root of unit in the complex field. By $m\equiv 2\pmod{4}$ , $\mathbb{F}_{3}^{*}\subset(\mathbb{F}_{3^{m}}^{*})^{2}$ ; by $-\frac{3^{m}-19}{5}\in C_{n-\delta_{3}}$ , $(\frac{3^{m}-19}{5},3^{m}-1)=2$ . For $a\in\mathbb{F}_{3}$ and $b\in\mathbb{F}_{3^{m}}$ ,

$\displaystyle W_{H}(c(a,b))$	$\displaystyle=$	$\displaystyle n-\frac{1}{3}\sum_{y\in\mathbb{F}_{3}}\sum_{i=0}^{n-1}\omega^{y[a(-1)^{i}+\operatorname{Tr}_{3^{m}/3}(b\alpha^{\delta_{3}i})]}$
	$\displaystyle=$	$\displaystyle\frac{2n}{3}-\frac{1}{3}\sum_{y\in\mathbb{F}_{3}^{*}}\sum_{i=0}^{n-1}\omega^{y[a(-1)^{i}+\operatorname{Tr}_{3^{m}/3}(b\alpha^{\delta_{3}i})]}$
	$\displaystyle=$	$\displaystyle\frac{2n}{3}-\frac{1}{3}\sum_{y\in\mathbb{F}_{3}^{}}\omega^{ya}\sum_{x\in\mathbb{F}_{3^{m}}^{}}\omega^{\operatorname{Tr}_{3^{m}/3}(bx^{2})}.$

Suppose that $a=0$ and $b=0$ . Then $W_{H}(c(a,b))=0$ .

Suppose that $a\neq 0$ and $b=0$ . Then

\displaystyle W_{H}(c(a,b))

\displaystyle=

\displaystyle\frac{2n}{3}-\frac{n}{3}\sum_{y\in\mathbb{F}_{3}^{*}}\omega^{ya}=n.

Suppose that $a=0$ and $b\neq 0$ . Then

			$\displaystyle W_{H}(c(a,b))=\frac{2n}{3}-\frac{2}{3}(\sum_{x\in\mathbb{F}_{3^{m}}}\omega^{\operatorname{Tr}_{3^{m}/3}(bx^{2})}-1)=2\cdot 3^{m-1}-\frac{2}{3}\eta(b)G(\eta)$
		$\displaystyle=$	$\displaystyle\left\{\begin{array}[]{ll}\frac{2}{3}\cdot(3^{m}-3^{\frac{m}{2}}),&\mbox{ if }b\mbox{ is a square},\\ \frac{2}{3}\cdot(3^{m}+3^{\frac{m}{2}}),&\mbox{ if }b\mbox{ is not a square}.\end{array}\right.$

Suppose that $a\neq 0$ and $b\neq 0$ . Then

			$\displaystyle W_{H}(c(a,b))=\frac{2n}{3}-\frac{1}{3}\sum_{y\in\mathbb{F}_{3}^{}}\omega^{ya}\sum_{x\in\mathbb{F}_{3^{m}}^{}}\omega^{\operatorname{Tr}_{3^{m}/3}(bx^{2})}$
		$\displaystyle=$	$\displaystyle\frac{2n}{3}-\frac{1}{3}(-1)\sum_{x\in\mathbb{F}_{3^{m}}}(\chi(\operatorname{Tr}_{3^{m}/3}(bx^{2}))-1)$
		$\displaystyle=$	$\displaystyle\frac{2n}{3}-\frac{1}{3}+\frac{1}{3}\eta(b)G(\eta)$
		$\displaystyle=$	$\displaystyle\left\{\begin{array}[]{ll}\frac{1}{3}(2\cdot 3^{m}+3^{\frac{m}{2}})-1,&\mbox{ if }b\mbox{ is a square },\\ \frac{1}{3}(2\cdot 3^{m}-3^{\frac{m}{2}})-1,&\mbox{ if }b\mbox{ is not a square. }\end{array}\right.$

Note that it is easy to obtain their frequencies and this completes the proof. ∎

Example 3.

Let $p=3$ , $m=6$ , and $n=p^{m}-1=728$ . Then the BCH code in Theorem 4.3 has weight enumerator

1+364z^{468}+728z^{476}+728z^{494}+364z^{504}+2z^{728},

which is confirmed by Magma.

4.2. Ternary LCD BCH Codes

Let $n=3^{m}-1$ and $\alpha$ a primitive element of $\mathbb{F}_{3^{m}}$ . Define a ternary LCD BCH code $\mathcal{C}_{(3,n,-t,2t)}=\langle g(x)\rangle$ , where $t$ is a positive integer, $Z=\bigcup_{|i|<t}C_{i}$ is a defining set, and $g(x)=\prod_{i\in Z}(x-\alpha^{i})$ . Now we shall choose some $t$ to compute the weight distributions of the ternary LCD BCH cyclic codes.

Theorem 4.4.

Let $m$ be an integer and $\delta_{1}=\frac{3^{m}-1}{2}$ . Then

\mathcal{C}_{(3,3^{m}-1,2\delta_{1},-\delta_{1})}=\{c(a)=(\operatorname{Tr}_{3^{m}/3}(ax))_{x\in\mathbb{F}_{3^{m}}^{*}}:a\in\mathbb{F}_{3^{m}}\}

is a ternary LCD BCH cyclic code with parameters $[3^{m}-1,1,3^{m}-1]$ and its designed distance $3^{m}-1$ .

Proof.

By Theorem 3.1, $\delta_{1}$ is the largest abstract coset leader, the parity-check polynomial of $\mathcal{C}_{(3,3^{m}-1,2\delta_{1},-\delta_{1})}$ is $h(x)=\frac{x^{n}-1}{g(x)}=x+1$ , where $h(x)$ is irreducible over $\mathbb{F}_{3}$ , if $\alpha$ is an $n$ th root of unit in $\mathbb{F}_{3^{m}}$ , $h(\alpha^{\delta_{1}})=0$ , $\deg(h(x))=1$ , and $h(x)$ is a self-reciprocal polynomial.

Let $\beta=\alpha^{\delta_{1}}$ . Then

\mathcal{C}_{(3,3^{m}-1,2\delta_{1},-\delta_{1})}=\{c(a)=(a\beta^{i})_{i=0}^{n-1}:a\in\mathbb{F}_{3}\}.

So, it has parameters $[3^{m}-1,1,3^{m}-1]$ and it has one all zeros codeword and two codewords with weight $3^{m}-1$ . ∎

Theorem 4.5.

Let $m$ be an odd integer, $\delta_{1}=\frac{3^{m}-1}{2}$ , $\delta_{2}=\frac{3^{m-1}-1}{4}+3^{m-2}$ , $Z=(\bigcup_{|s|<\delta_{2}}C_{s})\bigcup C_{\delta_{1}}$ , and $g(x)=\prod_{i\in Z}(x-\alpha^{i})$ . Then

\mathcal{C}_{(3,3^{m}-1,2\delta_{2},-\delta_{2})}=\{c(a,b)=(\operatorname{Tr}_{3^{m}/3}(ax+bx^{-1}))_{x\in\mathbb{F}_{3^{m}}^{*}}:a,b\in\mathbb{F}_{3^{m}}\}

is a ternary LCD cyclic code with parameters $[3^{m}-1,2m,\geq 2\delta_{2}]$ and its designed distance $2\delta_{2}$ .

Proof.

By Theorem 3.2, $\delta_{2}$ is the second largest absolute coset leader, the parity-check polynomial of $\mathcal{C}_{(3,3^{m}-1,2\delta_{2},-\delta_{2})}$ is $h(x)=\frac{x^{n}-1}{g(x)}=f(x)\widehat{f}(x)$ , where $f(x)$ is irreducible over $\mathbb{F}_{3}$ , $f(\alpha^{\delta_{2}})=0$ , $\deg(f(x))=m$ , and $\widehat{f}(x)$ is a reciprocal polynomial of $f(x)$ .

Let $\beta=\alpha^{\delta_{2}}$ . Then by Delsarte’s Theorem [5],

\mathcal{C}_{(3,3^{m}-1,2\delta_{2},-\delta_{2})}=\{c(a,b)=(\operatorname{Tr}_{3^{m}/3}(a\beta^{i}+b(\beta^{-1})^{i}))_{i=0}^{n-1}:a,b\in\mathbb{F}_{3^{m}}\}.

On the other hand, by $m$ odd, $-\frac{3^{m}+1}{4}\in C_{\delta_{2}}$ and $\gcd(4,3^{m}+1)=1$ , we get that $\gcd(\delta_{2},3^{m}-1)=1$ and $\beta$ is a primitive element of $\mathbb{F}_{3^{m}}$ . Hence

\mathcal{C}_{(3,3^{m}-1,2\delta_{2},-\delta_{2})}=\{c(a,b)=(\operatorname{Tr}_{3^{m}/3}(ax+bx^{-1}))_{x\in\mathbb{F}^{*}_{3^{m}}}:a,b\in\mathbb{F}_{3^{m}}\}.

By Theorem 3.1 and BCH bound, it has parameters $[3^{m}-1,2m,\geq 2\delta_{2}]$ . ∎

Let $a,b\in\mathbb{F}_{3^{m}}$ , the Kloosterman sum $K_{m}(a,b)$ is defined over $\mathbb{F}_{3^{m}}$ as follows:

K_{m}(a,b)=\sum_{x\in\mathbb{F}^{*}_{3^{m}}}\chi(ax+bx^{-1}),

where $\chi$ is the canonical additive character of $\mathbb{F}_{3^{m}}$ .

Corollary 1.

Let $m$ be an odd integer. Then for $a,b\in\mathbb{F}_{3^{m}}$ and $(a,b)\neq(0,0)$ ,

K_{m}(a,b)\leq\frac{3^{m}+2\cdot 3^{m-1}-1}{4}.

Proof.

For $a,b\in\mathbb{F}_{3^{m}}$ and $(a,b)\neq(0,0)$ , by Theorem 4.5,

$\displaystyle W_{H}(c(a,b))$	$\displaystyle=$	$\displaystyle n-\|\{x\in\mathbb{F}_{3^{m}}^{*}:\operatorname{Tr}_{3^{m}/3}(ax+bx^{-1})=0\}\|$
	$\displaystyle=$	$\displaystyle n-\frac{1}{3}\sum_{y\in\mathbb{F}_{3}}\sum_{x\in\mathbb{F}_{3^{m}}^{*}}\chi(y(ax+bx^{-1}))$
	$\displaystyle=$	$\displaystyle\frac{2n}{3}-\frac{2}{3}K_{m}(a,b).$

Hence $\frac{2n}{3}-\frac{2}{3}K_{m}(a,b)\geq 2(\frac{3^{m-1}-1}{4}+3^{m-2})$ and $K_{m}(a,b)\leq\frac{3^{m}+2\cdot 3^{m-1}-1}{4}$ . ∎

Remark.

Numerical examples by Magma show that the bound here is not tight in general.

Theorem 4.6.

Let $m$ be an even integer, $\delta_{1}=\frac{3^{m}-1}{2}$ , $\delta_{2}=\frac{3^{m}-1}{4}$ , $Z=\bigcup_{|s|<\delta_{2}}C_{s}$ , and $g(x)=\prod_{i\in Z}(x-\alpha^{i})$ . Then

\mathcal{C}_{(3,3^{m}-1,2\delta_{2},-\delta_{2})}=\{c(a,b)=(a\alpha^{\delta_{1}i}+\operatorname{Tr}_{3^{2}/3}(b\alpha^{\delta_{2}i}))_{i=0}^{n-1}:a\in\mathbb{F}_{3},b\in\mathbb{F}_{3^{2}}\}

is a ternary LCD BCH code with parameters $[3^{m}-1,3,\frac{1}{2}\cdot(3^{m}-1)]$ and the weight distribution in Table 3.

Table 3
Weight Frequency 0 1 $\frac{1}{2}\cdot(3^{m}-1)$ 12 $\frac{3}{4}\cdot(3^{m}-1)$ 8 $3^{m}-1$ 6

Proof.

By Theorem 3.2, $\delta_{2}$ is the second largest abstract coset leader and the parity-check polynomial of $\mathcal{C}_{(3,3^{m}-1,2\delta_{2},-\delta_{2})}$ is $h(x)=\frac{x^{n}-1}{g(x)}=f_{1}(x)f_{2}(x)$ , where $f_{1}(x)$ and $f_{2}(x)$ are irreducible over $\mathbb{F}_{3}$ , $f_{1}(x)=x+1$ , $f_{1}(\delta_{1})=0$ , $f_{2}(x)=x^{2}+1$ , and $f_{2}(\delta_{2})=0$ .

Let $\zeta_{4}=\alpha^{\delta_{2}}\in\mathbb{F}_{3^{2}}$ be a $4$ -th primitive root of unit and $\alpha^{\delta_{1}}=-1$ . Then by Delsarte’s Theorem [5],

\mathcal{C}_{(3,3^{m}-1,-\delta_{2},2\delta_{2})}=\{c(a)=(a(-1)^{i})+\operatorname{Tr}_{3^{2}/3}(b\zeta_{4}^{i}))_{i=0}^{n-1}:a\in\mathbb{F}_{3},b\in\mathbb{F}_{3^{2}}\}.

Let $\omega$ be a 3-th primitive root of unit in the complex field. By $m\equiv 0\pmod{4}$ , $8|3^{m}-1$ and $\mathbb{F}_{3}^{*}\subset(\mathbb{F}_{3^{2}}^{*})^{2}$ . Denote $Z(c(a,b))=|i\in\{0,1,\ldots,n-1\}:a(-1)^{i}+\operatorname{Tr}_{3^{2}/3}(b\zeta_{4}^{i})=0|.$ Then

			$\displaystyle W_{H}(c(a,b))=n-Z(c(a,b))$
		$\displaystyle=$	$\displaystyle n-\frac{1}{3}\sum_{y\in\mathbb{F}_{3}}\sum_{i=0}^{n-1}\omega^{y(a(-1)^{i})+\operatorname{Tr}_{3^{2}/3}(b\zeta_{4}^{i}))}$
		$\displaystyle=$	$\displaystyle\frac{2n}{3}-\frac{n}{12}\sum_{y\in\mathbb{F}_{3}^{*}}\omega^{ay}\sum_{i=0}^{3}\omega^{\operatorname{Tr}_{3^{2}/3}(by(-\zeta_{4})^{i}))}$
		$\displaystyle=$	$\displaystyle\frac{2n}{3}-\frac{n}{24}\sum_{y\in\mathbb{F}_{3}^{}}\omega^{ay}\sum_{x\in\mathbb{F}_{3^{2}}^{}}\omega^{\operatorname{Tr}_{3^{2}/3}(bx^{2})}.$

Note that $(\mathbb{F}_{3^{2}}^{*})^{2}=\langle\zeta_{4}\rangle$ and $\mathbb{F}_{3}^{*}\subset(\mathbb{F}_{3^{2}}^{*})^{2}$ .

Suppose that $a=0$ and $b=0$ . Then $W_{H}(c(a,b))=0$ .

Suppose that $a=0$ and $b\neq 0$ . Then by Lemma 2.3,

$\displaystyle W_{H}(c(a,b))$	$\displaystyle=$	$\displaystyle\frac{2n}{3}-\frac{n}{12}(\sum_{x\in\mathbb{F}_{9}}\omega^{y(\operatorname{Tr}_{3^{2}/3}(bx^{2}))}-1)=\frac{2n}{3}-\frac{n}{12}(\eta^{\prime}(b)G(\eta^{\prime})-1)$
	$\displaystyle=$	$\displaystyle\frac{3n}{4}-\frac{n}{12}\eta^{\prime}(b)G(\eta^{\prime})$
	$\displaystyle=$	$\displaystyle\left\{\begin{array}[]{ll}\frac{n}{2},&\mbox{ if }b\mbox{ is a square, }\\ n,&\mbox{ if }b\mbox{ is not a square. }\end{array}\right.$

where $\eta^{\prime}$ is a multiplicative character of order $2$ in $\mathbb{F}_{9}$ .

Suppose that $a\neq 0$ and $b=0$ . Then

\displaystyle W_{H}(c(a,b))=\frac{2n}{3}-\frac{n}{24}\sum_{y\in\mathbb{F}_{3}^{*}}\omega^{ay}(3^{2}-1)=n.

Suppose that $a\neq 0$ and $b\neq 0$ . By Lemma 2.3,

$\displaystyle W_{H}(c(a,b))$	$\displaystyle=$	$\displaystyle\frac{2n}{3}-\frac{n}{24}(-1)(\sum_{x\in\mathbb{F}_{9}}\omega^{y(\operatorname{Tr}_{3^{2}/3}(bx^{2}))}-1)$
	$\displaystyle=$	$\displaystyle\frac{2n}{3}+\frac{n}{24}(\eta^{\prime}(b)G(\eta^{\prime})-1)=\frac{3n}{4}+\frac{n}{24}\eta^{\prime}(b)G(\eta^{\prime})$
	$\displaystyle=$	$\displaystyle\left\{\begin{array}[]{ll}\frac{3n}{4},&\mbox{ if }b\mbox{ is a square, }\\ \frac{n}{2},&\mbox{ if }b\mbox{ is not a square. }\end{array}\right.$

Note that it is easy to obtain their frequencies and this completes the proof. ∎

Example 4.

Let $p=3$ , $m=4$ , and $n=p^{m}-1=81$ . Then the LCD BCH code in Theorem 4.6 has weight enumerator $1+12z^{40}+8z^{60}+6z^{80},$ which is confirmed by Magma.

Theorem 4.7.

Let $m\equiv 2\pmod{4}$ , $\delta_{1}=\frac{3^{m}-1}{2}$ , $\delta_{2}=\frac{3^{m}-1}{4}$ $\delta_{3}=\frac{3^{m-6}-1}{5}+3^{m-6}+2\cdot 3^{m-5}+2\cdot 3^{m-3}+3^{m-2}$ , $Z=(\bigcup_{|s|<\delta_{3}}C_{s})\bigcup C_{\delta_{1}}\bigcup C_{\delta_{2}}$ , $g(x)=\prod_{i\in Z}(x-\alpha^{i})$ . Then

\mathcal{C}_{(3,3^{m}-1,2\delta_{3},-\delta_{3})}=\{c(a,b)=(\operatorname{Tr}_{3^{m}/3}(ax^{2}+bx^{-2}))_{x\in\mathbb{F}_{3^{m}}^{*}}:a,b\in\mathbb{F}_{3^{m}}\}

is a ternary LCD BCH code with parameters $[3^{m}-1,2m,\geq 2\delta_{3}]$ and its designed distance $2\delta_{3}$ .

Proof.

By Theorem 3.3, $\delta_{3}$ is the third largest abstract coset leader, the parity-check polynomial of $\mathcal{C}_{(3,3^{m}-1,-\delta_{3},2\delta_{3})}$ is $h(x)=\frac{x^{n}-1}{g(x)}=f(x)\widehat{f}(x)$ , where $f(x)$ is irreducible over $\mathbb{F}_{3}$ , $f(\alpha^{\delta_{3}})=0$ , $\deg(f(x))=m$ , and $\widehat{f}(x)$ is a reciprocal polynomial of $f(x)$ .

Let $\beta=\alpha^{\delta_{3}}$ . Then by Delsarte’s Theorem [5],

\mathcal{C}_{(3,3^{m}-1,2\delta_{3},-\delta_{3})}=\{c(a,b)=(\operatorname{Tr}_{3^{m}/3}(a\beta^{i}+b(\beta^{-1})^{i}))_{i=0}^{n-1}:a,b\in\mathbb{F}_{3^{m}}\}.

On the other hand, by $m\equiv 2\pmod{4}$ , $\frac{3^{m}-19}{5}\in C_{\delta_{3}}$ , $\gcd(5,3^{m}-1)=1$ , and $\gcd(3^{m}-19,3^{m}-1)=\gcd(18,3^{m}-1)=2$ , we get that $\gcd(\delta_{3},3^{m}-1)=2$ and $\beta$ is a semi-primitive element of $\mathbb{F}_{3^{m}}$ . Hence

\mathcal{C}_{(3,3^{m}-1,2\delta_{3},-\delta_{3})}=\{c(a,b)=(\operatorname{Tr}_{3^{m}/3}(ax^{2}+bx^{-2}))_{x\in\mathbb{F}^{*}_{3^{m}}}:a,b\in\mathbb{F}_{3^{m}}\}.

By Theorem 3.1 and BCH bound, the code has parameters $[3^{m}-1,2m,\geq 2\delta_{3}]$ . ∎

5. Concluding remarks

In this paper, several classes of ternary primitive BCH codes and LCD BCH codes were studied according to the first, second and third largest absolute coset leaders. The weight distributions of these codes were given except two of them, whose weight distributions rely on the calculation of Kloosterman sums.

Acknowledgments

The authors are very grateful to the reviewers and the Editor for their valuable suggestions that improved the quality of this paper.

References

[1] C. Carlet and S. Guilley, \doititleComplementary dual codes for countermeasures to side-channel attacks, Adv. Math. Commun., 10(1)(2016), 131–150.
[2] C. Carlet, S. Mesnager, C. Tang and Y. Qi, \doititleEuclidean and Hermitian LCD MDS codes, Des. Codes Cryptogr., 86(2018), 2605–2618.
[3] C. Carlet, S. Mesnager, C. Tang, Y. Qi and R. Pellikaan, \doititleLinear codes over $\mathbb{F}_{q}$ which are equivalent to LCD codes, IEEE Trans. Inf. Theory, 64(2018), 3010–3017.
[4] B. Chen and H. Liu, \doititleNew constructions of MDS codes with complementary duals, IEEE Trans. Inf. Theory, 63(2017), 2843–2847.
[5] P. Delsarte, \doititleOn subfield subcodes of modified Reed-Solomon codes, IEEE Trans. Inf. Theory, 21(1975), 575–576.
[6] C. Ding, \doititleParameters of several classes of BCH codes, IEEE Trans. Inf. Theory, 61(2015), 5322–5330.
[7] C. Ding, C. Fan and Z. Zhou, \doititleThe dimension and minimum distance of two classes of primitive BCH codes, Finite Fields Appl., 45(2017), 237–263.
[8] X. Huang, Q. Yue, Y. Wu and X. Shi, \doititleBinary Primitive LCD BCH codes, Des. Codes Cryptogr., 88(2020), 2453–2473.
[9] L. Jin, \doititleConstruction of MDS codes with complementary duals, IEEE Trans. Inf. Theory, 63(2017), 2843–2847.
[10] C. Li, \doititleHermitian LCD codes from cyclic codes, Des. Codes Cryptogr., 86(2018), 2261–2278.
[11] C. Li, C. Ding and S. Li, \doititleLCD cyclic codes over finite fields, IEEE Trans. Inf. Theory, 63(2017), 4344–4356.
[12] C. Li, P. Wu and F. Liu, \doititleOn two classes of primitive BCH codes and some related codes, IEEE Trans. Inf. Theory, 65(2019), 3830–3840.
[13] C. Li, Q. Yue and F. Li, \doititleWeight distributions of cyclic codes with respect to pairwise coprime order elements, Finite Fields Appl., 28(2014), 94–114.
[14] R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley Publishing Inc.,1983.
[15] F. Li, Q. Yue and Y. Wu, \doititleDesigned distances and parameters of new LCD BCH codes over finite fields, Cryptogr. Commun., 12(2020), 147–163.
[16] S. Li, C. Ding, M. Xiong and G. Ge, \doititleNarrow-sense BCH codes over GF $(q)$ with length $n=\frac{q^{m}-1}{q-1}$ , IEEE Trans. Inf. Theory, 63(2017), 7219-7236.
[17] S. Li , C. Li, C. Ding and H. Liu, \doititleTwo families of LCD BCH codes, IEEE Trans. Inf. Theory, 63( 2017), 5699–5717.
[18] H. Liu, C. Ding and C. Li, \doititleDimensions of three types of BCH codes over GF $(q)$ , Discrete Math., 340(2017), 1910–1927.
[19] J. L. Massey, \doititleReversible codes, Information and Control, 7(1964), 369–380.
[20] X. Shi, Q. Yue and S. Yang, \doititleNew LCD MDS codes constructed from generalized Reed-Solomon codes, J. Algebra Appl., 18(2018), 1950150.
[21] K. K. Tzeng and C. R. P. Hartmann, \doititleOn the minimum distance of certain reversible cyclic codes, IEEE Trans. Inf. Theory, 16(1970),644–646.
[22] Y. Wu and Q. Yue, \doititleFactorizations of binomial polynomials and enumerations of LCD and self-dual constacyclic codes, IEEE Trans. Inf. Theory, 65(2019),1740–1751.
[23] H. Yan, H. Liu, C. Li and S. Yang, \doititleParameters of LCD BCH codes with two lengths, Adv. Math. Commun., 12(2018), 579–594,.
[24] X. Yang and J. L. Massey, \doititleThe necessary and sufficient condition for a cyclic code to have a complementary dual, Discrete Math., 126(1994), 391–393.
[25] Z. Zhou, X. Li, C. Tang and C. Ding, \doititleBinary LCD codes and self-orthogonal codes from a generic construction, IEEE Trans. Inf. Theory, 65(2019), 16–27.

Received March 2021; 1st revision ; final revision .

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