A class of narrow-sense BCH codes over $\mathbb{F}_q$ of length $\frac{q^m-1}{2}$

TL;DR

This paper characterizes a class of narrow-sense BCH codes over finite fields with length (q^m-1)/2, determining their weight enumerators, minimal distances, and trace representations using association schemes and cyclotomic cosets.

Contribution

It provides explicit trace representations and weight enumerators for a new class of BCH codes of length (q^m-1)/2, linking cyclotomic coset leaders to code parameters.

Findings

Determined weight enumerators of the codes.

Proved minimal and Bose distances equal to designed distances.

Established trace representations for the codes.

Abstract

BCH codes with efficient encoding and decoding algorithms have many applications in communications, cryptography and combinatorics design. This paper studies a class of linear codes of length $ \frac{q^m-1}{2}$ over $\mathbb{F}_q$ with special trace representation, where $q$ is an odd prime power. With the help of the inner distributions of some subsets of association schemes from bilinear forms associated with quadratic forms, we determine the weight enumerators of these codes. From determining some cyclotomic coset leaders $\delta_i$ of cyclotomic cosets modulo $ \frac{q^m-1}{2}$, we prove that narrow-sense BCH codes of length $ \frac{q^m-1}{2}$ with designed distance $\delta_i=\frac{q^m-q^{m-1}}{2}-1-\frac{q^{ \lfloor \frac{m-3}{2} \rfloor+i}-1}{2}$ have the corresponding trace representation, and have the minimal distance $d=\delta_i$ and the Bose distance $d_B=\delta_i$, where $1\leq i\leq \lfloor \frac{m+3}{4} \rfloor$.

Equations

$${\mathcal{C}}_{1,h}=\left\{{\left({\mathrm{Tr}}_1^m\left(\sum _{j=h}^{\frac{m-1}{2}}{a}_j{\alpha }^{\left(q^j+1\right)l}\right)+a\right)}_{l=0}^{n-1}:a_h,\dots ,a_{\frac{m-1}{2}}\in {\mathbb{F}}_{q^m},a\in {\mathbb{F}}_q\right\}$$

$${\mathcal{C}}_{2,h}=\left\{{\left({\mathrm{Tr}}_1^m\left(a_{\frac{m}{2}}{\alpha }^{\left(q^{\frac{m}{2}}+1\right)l}+\sum _{j=h}^{\frac{m-2}{2}}{a}_j{\alpha }^{\left(q^j+1\right)l}\right)+a\right)}_{l=0}^{n-1}:a_h,\dots ,a_{\frac{m-2}{2}}\in {\mathbb{F}}_{q^m},a_{\frac{m}{2}}\in {\mathbb{F}}_{q^{\frac{m}{2}}},a\in {\mathbb{F}}_q\right\}.$$

$$\psi :{\mathbb{F}}_q^n$$

$$(c_0,c_1,\dots ,c_{n-1})$$

$$g_{(n,q,m,\delta )}(x)=lcm(m_1(x),\dots ,m_{\delta -1}(x))\text{and}{\tilde{g}}_{(n,q,m,\delta )}(x)=(x-1)g_{(n,q,m,\delta )}(x),$$

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

$$m_s(x)=\prod _{i\in C_s}(x-{\beta }^i)\in {\mathbb{F}}_q[x],$$

$$x^n-1=\prod _{s\in {\Gamma }_{(n,q)}}{m}_s(x).$$

$$h(x)=(x-1)\prod _{s\ge \delta ,s\in {\Gamma }_{(n,q)}}{m}_s(x),$$

$$\sum _{s\ge \delta ,s\in {\Gamma }_{(n,q)}}{l}_s+1.$$

$$\mathcal{C}=\{c(a_1,\dots ,a_s):a_j\in {\mathbb{F}}_{q^{l_j}},\text{for}1\le j\le s\},$$

$$N(b)=\left\{\begin{array}[]{ll}q^{n-1}+\tau\eta(-1)^{\frac{r-1}{2}}\eta(b)q^{n-\frac{r+1}{2}},&\hbox{if $r$ is odd;}\\ q^{n-1}+\tau\eta(-1)^{\frac{r}{2}}v(b)q^{n-\frac{r+2}{2}},&\hbox{if $r$ is even.}\end{array}\right.$$

$$A_i=\sum _{k=0}^n{P}_i(k)E_k\text{and}{E}_k=\frac{1}{|X|}\sum _{i=0}^n{Q}_k(i)A_i.$$

$${\left(B({\alpha }_i,{\alpha }_j)\right)}_{1\le i,j\le m}.$$

$$R_{r,\tau }=\{(A,B)\in X(m,q)\times X(m,q):A-B\in X_{r,\tau }\}.$$

$$a_{r,\tau }=\frac{|(Y\times Y)\cap R_{r,\tau }|}{|Y|}.$$

$$a_{k,ϵ}^{\prime }=\sum _{r,\tau }{Q}_{k,ϵ}(r,\tau )a_{r,\tau }.$$

$$\left[\begin{array}{c}n\\ k\end{array}\right]=\prod _{i=1}^k\frac{q^{2n-2i+2}-1}{q^{2i}-1}$$

$$a_{2i-1,\tau }$$

$$a_{2i,\tau }$$

$$a_{2i-1,\tau }=$$

$$a_{2i,\tau }=$$

$$+\frac{\tau }{2}\eta (-1{)}^i{q}^i\left[\begin{array}{c}n\\ i\end{array}\right]\sum _{j=0}^{i-\delta }(-1{)}^j{q}^{j(j-1)}\left[\begin{array}{c}i\\ j\end{array}\right](\frac{|Y|}{q^{(2n-1)(n+j-i)}{q}^{2n}}-1)$$

$$a_{2i-1,\tau }=$$

$$a_{2i,\tau }=$$

$$+\frac{\tau }{2}\eta (-1{)}^i{q}^i\left[\begin{array}{c}n\\ i\end{array}\right]\sum _{j=0}^{i-\delta }(-1{)}^j{q}^{j(j-1)}\left[\begin{array}{c}i\\ j\end{array}\right](\frac{|Y|}{q^{(2n-1)(n+j-i)}{q}^{2n}}-1)$$

$$a_{2i-1,\tau }=$$

$$+\frac{1}{2}(-1{)}^{i-\delta }{q}^{(i-\delta )(i-\delta -1)}\left[\begin{array}{c}n\\ \delta -1\end{array}\right](\frac{|Y|}{q^{(2n+1)(n-\delta +1)}}-1)\left(\left[\begin{array}{c}n-\delta \\ n-i+1\end{array}\right](q^{n-\delta +1}+1)-\left[\begin{array}{c}n-\delta +1\\ n-i+1\end{array}\right]\right)$$

$$a_{2i,\tau }=$$

$$+\frac{1}{2}\eta (-1{)}^{i-\delta }{q}^{(i-\delta +1)(i-\delta )}\left[\begin{array}{c}n\\ \delta -1\end{array}\right]\left[\begin{array}{c}n-\delta \\ n-i\end{array}\right](q^{n-\delta +1}+1)(\frac{|Y|}{q^{(2n+1)(n-\delta +1)}}-1)$$

$${\mathcal{C}}_{1,h}=\left\{{\left({\mathrm{Tr}}_1^m\left(\sum _{j=h}^{\frac{m-1}{2}}{a}_j{\alpha }^{\left(q^j+1\right)l}\right)+a\right)}_{l=0}^{n-1}:a_h,\dots ,a_{\frac{m-1}{2}}\in {\mathbb{F}}_{q^m},a\in {\mathbb{F}}_q\right\}$$

$${\tilde{\mathcal{C}}}_{1,h}=\left\{{\left({\mathrm{Tr}}_1^m\left(\sum _{j=h}^{\frac{m-1}{2}}{a}_j{\alpha }^{\left(q^j+1\right)l}\right)\right)}_{l=0}^{n-1}:a_h,\dots ,a_{\frac{m-1}{2}}\in {\mathbb{F}}_{q^m},a\in {\mathbb{F}}_q\right\}.$$

$${\mathcal{C}}_{2,h}=\left\{{\left({\mathrm{Tr}}_1^m\left(a_{\frac{m}{2}}{\alpha }^{\left(q^{\frac{m}{2}}+1\right)l}+\sum _{j=h}^{\frac{m-2}{2}}{a}_j{\alpha }^{\left(q^j+1\right)l}\right)+a\right)}_{l=0}^{n-1}:a_h,\dots ,a_{\frac{m-2}{2}}\in {\mathbb{F}}_{q^m},a_{\frac{m}{2}}\in {\mathbb{F}}_{q^{\frac{m}{2}}},a\in {\mathbb{F}}_q\right\}$$

$${\tilde{\mathcal{C}}}_{2,h}=\left\{{\left({\mathrm{Tr}}_1^m\left(a_{\frac{m}{2}}{\alpha }^{\left(q^{\frac{m}{2}}+1\right)l}+\sum _{j=h}^{\frac{m-2}{2}}{a}_j{\alpha }^{\left(q^j+1\right)l}\right)\right)}_{l=0}^{n-1}:a_h,\dots ,a_{\frac{m-2}{2}}\in {\mathbb{F}}_{q^m},a_{\frac{m}{2}}\in {\mathbb{F}}_{q^{\frac{m}{2}}},a\in {\mathbb{F}}_q\right\}.$$

$$Q(x)={\mathrm{Tr}}_1^m\left(\sum _{j=1}^l{a}_j{x}^{q^{k_j}+1}\right),$$

Full text

1 Introduction

As an important class of cyclic codes, BCH codes with efficient encoding and decoding algorithms have many applications in communications, cryptography and combinatorics design. They were independently discovered by Hocquenghem H59 and Bose, Ray-Chaudhuri BR60 , and were generalized from binary fields to finite fields by Gorenstein and Zierler GZ61 . The determination of parameters of BCH codes is an interesting and difficult problem. There are many results on BCH codes AKS07 ; B70 ; C90 ; C98 ; CHZ06 ; D15 ; D18 ; DDZ15 ; DFZ17 ; DFK97 ; FL08 ; FTKL86 ; HS73 ; K69 ; KT69 ; KL99 ; KL01 ; LDL-17 ; LLDL17 ; LDL17 ; MS77 ; M62 ; P69 ; W69 ; Y18 ; YF00 ; YH96 ; ZD14 .

Let $m$ be a positive integer and $\mathbb{F}_{q^{m}}$ be a finite field. Let $\beta$ be an element in $\mathbb{F}_{q^{m}}$ with order $n$, where $n\mid(q^{m}-1)$. A BCH code $\mathcal{C}$ over $\mathbb{F}_{q}$ with designed distance $\delta$ is a cyclic code with a generator polynomial $g(x)$, where $g(x)$ is determined by its $\delta-1$ consecutive roots $\beta^{l},\beta^{l+1},\ldots,\beta^{l+\delta-1}$. The minimal distance of the BCH code $\mathcal{C}$ is greater than $\delta$. When $n=q^{m}-1$, the code $\mathcal{C}$ is called primitive. When $l=1$, this code $\mathcal{C}$ is called narrow-sense. Two narrow-sense BCH codes with different designed distances may be the same. The Bose distance $d_{B}$ of a BCH code is the largest designed distance. The Bose distance satisfies that $d_{B}\leq d\leq d_{B}+4$ C98 , where $d$ is the minimum distance. The determination of the minimum distance $d$ and the Bose distance of a narrow-sense BCH code attracts much interest AKS07 ; C90 ; C98 ; DDZ15 ; KL72 ; KLP67 ; KT69 ; M80 ; M62 ; YF00 . Some results on $[n,k,d]$ narrow-sense BCH codes over $\mathbb{F}_{q}$ such that $d=d_{B}$ are listed:

• •

the primitive case: $n=q^{m}-1$

• –

$d=d_{B}=q^{i}-1$, where $1\leq i\leq m-1$ P69 ;

• –

$q=2$, $d=d_{B}=2^{m-1}-2^{i}-1$, where $\frac{m-2}{2}\leq i\leq m-\lfloor\frac{m}{3}\rfloor-1$ B70 ;

• –

$q=2$, $d=d_{B}=2^{m-1-s}-2^{m-1-i-s}$, where $1\leq i\leq m-s-2$ and $0\leq s\leq m-2i$ KL72 ;

• –

$d=d_{B}=q^{i}+1$, where $(m-i)\mid i$ or $2i\mid m$ DDZ15 ;

• –

$d=d_{B}=(q-l_{0})q^{m-l_{1}-1}-1$, where $0\leq l_{0}\leq q-2$ and $0\leq l_{1}\leq m-1$ D15 ;

• –

$d=d_{B}=q^{m}-q^{m-1}-q^{i}-1$, where $\frac{m-2}{2}\leq i\leq m-\lfloor\frac{m}{3}\rfloor-1$ [Theorem 1.2, L17 ].

• •

$n=\frac{q^{m}-1}{2}$

• –

$q=3$, $d=d_{B}=\frac{q^{m}-q^{m-1}}{2}-1-\frac{q^{\lfloor\frac{m-3}{2}\rfloor+i}-1}{2}$ for $i=1,2$ LDXG17 ;

• –

$d=d_{B}=\frac{q^{m}-q^{m-1}}{2}-1-\frac{q^{\lfloor\frac{m-3}{2}\rfloor+i}-1}{2}$ for $i=1,2$ ZSK19 ;

• –

$d=d_{B}=\frac{q^{m}-q^{m-1}}{2}-1-\frac{q^{\lfloor\frac{m-3}{2}\rfloor+i}-1}{2}$, where $1\leq i\leq\lfloor\frac{m+3}{4}\rfloor$ [this paper].

To determine the minimal distance $d$ and the Bose distance $d_{B}$ of narrow-sense primitive BCH codes with designed distance $q^{m}-q^{m-1}-q^{i}-1$, Li L17 employed the framework of association schemes from bilinear forms associated with quadratic forms and presented the weight enumerators of these BCH codes, where these quadratic forms are corresponding to the trace representation of BCH codes. In TDX19 , Tang et al. studied a class of ternary linear codes with trace representation, determined the weight distributions of some shortened codes and punctured codes of these three-weight subcodes, and presented some $2$-designs from these codes.

Motivated by these papers and their ideas, this paper studies the following linear codes $\mathcal{C}_{1,h}$ (resp. $\mathcal{C}_{2,h}$) of length $n=\frac{q^{m}-1}{2}$ with special trace representation over finite field $\mathbb{F}_{q}$ for $m$ odd (resp. even), where $q$ is odd prime power and

[TABLE]

and

[TABLE]

These codes correspond to quadratic forms. With the help of the theory of association schemes from bilinear forms associated with quadratic forms, we determine the weight enumerators of these codes. Determining some largest cyclotomic coset leaders of cyclotomic cosets modulo $n=\frac{q^{m}-1}{2}$, we prove that a narrow-sense BCH code with designed distance $\delta_{i}=\frac{q^{m}-q^{m-1}}{2}-1-\frac{q^{\lfloor\frac{m-3}{2}\rfloor+i}-1}{2}$ has the above trace representation and has the minimal distance $d$ such that $d=d_{B}=\delta_{i}$, where $1\leq i\leq\lfloor\frac{m+3}{4}\rfloor$.

The rest of the paper is organized as follows. Section 2 introduces some basic results on BCH codes, quadratic forms and association schemes. Section 3 studies some linear codes with trace representation, uses association schemes to determine the weight enumerators of these codes, and presents the minimum distance $d$ and the Bose distance of the narrow-sense BCH codes of length $n=\frac{q^{m}-1}{2}$ with designed distance $\delta_{i}$. Section 4 makes a conclusion.

2 Preliminaries

In this section, some results on BCH codes, quadratic forms and association schemes are introduced.

2.1 BCH codes

Let $q$ be a prime power. An $[n,k,d]$ linear code $\mathcal{C}$ over the finite field $\mathbb{F}_{q}$ is a $k$-dimensional subspace of $\mathbb{F}_{q}^{n}$, where $d$ is the minimal distance of $\mathcal{C}$. A linear code is called a cyclic code if $(c_{0},c_{1},\ldots,c_{n-1})\in\mathcal{C}$ implies that $(c_{n-1},c_{0},\ldots,c_{n-2})\in\mathcal{C}$. We also write a cyclic code as a principal ideal of the ring $\mathbb{F}_{q}[x]/(x^{n}-1)$, since there is a map

[TABLE]

Then $\mathcal{C}=\langle g(x)\rangle$, where $g(x)$ is a monic polynomial and is called the generator polynomial of $\mathcal{C}$. Note that $g(x)|(x^{n}-1)$. The polynomial $h(x)=g(x)|(x^{n}-1)$ is called the parity-check polynomial of $\mathcal{C}$. The code $\mathcal{C}$ is called a cyclic code with $s$ zeros (resp. $s$ nonzeros) if $g(x)$ (resp. $h(x)$) can be factored into a product of $s$ irreducible polynomials over $\mathbb{F}_{q}$.

Let $n$ be a positive integer and $m$ be the smallest positive integer such that $n|(q^{n}-1)$. Let $\alpha$ be a primitive element of $\mathbb{F}_{q^{m}}$ and $\beta=\alpha^{\frac{q^{m}-1}{n}}$. Let $m_{i}(x)$ be the minimal polynomial of $\beta^{i}$ over $\mathbb{F}_{q}$, where $0\leq i\leq n-1$. For each $2\leq\delta\leq n$, define two polynomials

[TABLE]

where $lcm$ denotes the least common multiple of the polynomials. Let $\mathcal{C}_{(n,q,m,\delta)}=\langle g_{(n,q,m,\delta)}(x)\rangle$ and $\tilde{\mathcal{C}}_{(n,q,m,\delta)}=\langle\tilde{g}_{(n,q,m,\delta)}(x)\rangle$. Then $\mathcal{C}_{(n,q,m,\delta)}$ is called a narrow-sense BCH code with designed distance $\delta$ and $\tilde{\mathcal{C}}_{(n,q,m,\delta)}$ is the even-like subcode of $\mathcal{C}_{(n,q,m,\delta)}$. Note that $dim(\tilde{\mathcal{C}}_{(n,q,m,\delta)})=dim(\mathcal{C}_{(n,q,m,\delta)})-1$ and the minimal distances of $\mathcal{C}_{(n,q,m,\delta)}$ and $\tilde{\mathcal{C}}_{(n,q,m,\delta)}$ are at least $\delta$ and $\delta+1$, respectively. If $n=q^{m}-1$, the code $\mathcal{C}_{(n,q,m,\delta)}$ is called a narrow-sense primitive BCH code. Note that two BCH codes with different designed distances may be the same. For a narrow-sense BCH code $\mathcal{C}_{(n,q,m,\delta)}$, the Bose distance MS77 of this code is defined by the largest designed distance $d_{B}=max(\delta^{\prime})$, where ${\mathcal{C}_{(n,q,m,\delta^{\prime})}=\mathcal{C}_{(n,q,m,\delta)}}$.

The narrow-sense BCH code $\mathcal{C}_{(n,q,m,\delta)}$ has close relation with $q$-cyclotomic cosets modulo $n$. The $q$-cyclotomic coset of $s$ modulo $n$ is defined by

[TABLE]

where $0\leq s\leq n-1$ and $l_{s}$ called the size of the $q$-cyclotomic coset is the smallest positive integer such that $s\equiv sq^{l_{s}}\mod n$. Note that $l_{s}\mid m$. The coset leader of $C_{s}$ is the smallest integer in $C_{s}$. Let $\Gamma_{n,q}$ be the set of all the coset leaders. Then $C_{s}\bigcap C_{t}=\emptyset$ and $\mathbb{Z}_{n}=\bigcup_{s\in\Gamma_{(n,q)}}C_{s}$, where $s,t\in\Gamma_{(n,q)}$ and $s\neq t$.

Let $m_{s}(x)$ be the minimal polynomial of $\beta^{s}$ over $\mathbb{F}_{q}$. Then

[TABLE]

$deg(m_{s}(x))=l_{s}$ and

[TABLE]

Hence, the generator polynomial $g_{(n,q,m,\delta)}(x)=\prod_{s\in\cup_{j=1}^{\delta-1}C_{j}}m_{s}(x)$, the parity-check polynomial of $\mathcal{C}_{(n,q,m,\delta)}$ is

[TABLE]

and the dimension of $\mathcal{C}_{(n,q,m,\delta)}$ is

[TABLE]

The following lemma gives the relation between the Bose distance $d_{B}$ of $\mathcal{C}_{(n,q,m,\delta)}$ and coset leaders.

Lemma 1 (Proposition 4, LDXG17 )

The Bose distance $d_{B}$ of the narrow-sense BCH code $\mathcal{C}_{(n,q,m,\delta)}$ is a coset leader of a $q$-cyclotomic coset modulo $n$. Furthermore, if $\delta$ is a coset leader, then $d_{B}=\delta$.

From Delsarte’s theorem D75 , a cyclic code has the following trace representation.

Proposition 1

Let $q$ be a prime power, $n$ be a positive integer, $m$ be the smallest positive integer such that $n|(q^{n}-1)$, and $\beta$ be an $n$-th primitive element of $\mathbb{F}_{q^{m}}$. Let $\mathcal{C}$ be a cyclic code of length $n$ over $\mathbb{F}_{q}$ with $s$ nonzeros. Let $\beta^{i_{1}},\ldots,\beta^{i_{s}}$ be $s$ roots of its parity-check polynomial $h(x)$, which are not conjugate with each other. Then the code $\mathcal{C}$ has the following trace representation

[TABLE]

where $l_{j}$ is the size of the $q$-cyclotomic coset $C_{i_{j}}$, $\mathrm{Tr}_{1}^{l_{j}}$ is the trace function from $\mathbb{F}_{q^{l_{j}}}$ to $\mathbb{F}_{q}$, and $c(a_{1},\ldots,a_{s})=\left(\sum_{j=1}^{s}\mathrm{Tr}_{1}^{l_{j}}(a_{j}\beta^{-li_{j}})\right)_{l=0}^{n-1}$.

2.2 Quadratic forms

This subsection introduces some results on quadratic forms LN97 . Let $q$ be an odd prime power and $V$ be an $n$-dimensional vector space over $\mathbb{F}_{q}$. A quadratic form $Q$ on $V$ is a function from $V$ to $\mathbb{F}_{q}$ satisfying $Q(\lambda x)=\lambda^{2}Q(x)$, where $x\in V$ and $\lambda\in\mathbb{F}_{q}$. A symmetric bilinear form $B_{Q}(x,y)=\frac{1}{2}(Q(x+y)-Q(x)-Q(y))$ on $V$ is associated with $Q$. The radical of $Q$ is $Rad(Q)=Q^{-1}(0)\cap Rad(B_{Q})$, where $Rad(B_{Q})=\{y\in V:B(x,y)=0,\forall x\in V\}$ is the radical of the symmetric bilinear form $B_{Q}$. The radical of $Q$ is a vector space over $\mathbb{F}_{q}$. The rank of $Q$ is $rank(Q)=n-dim(Rad(Q))$ and the rank of $B_{Q}$ is $rank(B_{Q})=n-dim(Rad(B_{Q}))$.

Lemma 2 (Lemma 3.6, L17 )

Let $q$ be an odd prime power, $Q$ be a quadratic form on the $n$-dimensional vector space $V$ over $\mathbb{F}_{q}$ and $B_{Q}$ be its associated bilinear form. Then $rank(Q)=rank(B_{Q})$.

Let $V=\mathbb{F}_{q}^{n}$ and $Q$ be a quadratic form on $V$. Then $Q$ has the following expression $Q(x)=\sum_{1\leq i,j\leq n}c_{ij}x_{i}x_{j}$. Two quadratic forms $Q$ and $Q^{\prime}$ are equivalent if there is an $n\times n$ nonsingular matrix $A$ such that $Q(x_{1},\ldots,x_{n})=Q^{\prime}((x_{1},\ldots,x_{n})A)$. Every quadratic form $Q$ is equivalent to $\sum_{i=1}^{r}a_{i}x_{i}^{2}$, where $a_{i}\in\mathbb{F}_{q}^{*}$ and $r=rank(Q)$. The type of $Q$ is $\tau=\eta(\prod_{i=1}^{r}a_{i})$, where $\eta$ is the quadratic character of $\mathbb{F}_{q}$.

Lemma 3 (Lemma 5.1, S15 )

Let $q$ be an odd prime power and $Q$ be quadratic form of rank $r$ and type $\tau$. Let $N(b)$ be the number of solutions $Q(x)=b$, where $h\in\mathbb{F}_{q}$. Then

[TABLE]

where $\eta(0)=0$, $v(x)=-1$ for $x\in\mathbb{F}_{q}^{*}$ and $v(0)=q-1$.

Let $V=\mathbb{F}_{q^{n}}$. Choose a basis $\alpha_{1},\ldots,\alpha_{n}$ of $V$ over $\mathbb{F}_{q}$. We have a bijection between $\mathbb{F}_{q^{n}}$ and $\mathbb{F}_{q}^{n}$. Note that the rank $r$ and the type $\tau$ of a quadratic form $Q$ on $V$ are independent of the choice of the basis.

2.3 Association schemes

This subsection introduces some results on association schemes MS77 ; HW93 ; WWMM03 . Let $X$ be a finite set and $R_{0},R_{1},\ldots,R_{n}$ be a partition of $X\times X$. Then a pair $\left(X,(R_{i})\right)$ is called an association scheme with $n$ classes if it satisfies

• •

$R_{0}=\{(x,x):x\in X\}$;

• •

for each $i$, there exists $j$ such that the inverse of $R_{i}$ equals $R_{j}$;

• •

if $(x,y)\in R_{k}$, the number of the set $\{z\in X:(x,z)\in R_{i},(z,y)\in R_{j}\}$ is a constant $p_{ij}^{k}$ depending on only $i$, $j$, and $k$, but not on the particular choice of $x$ and $y$.

An association scheme $\left(X,(R_{i})\right)$ is symmetric if for all $i$ the inverse of $R_{i}$ equals $R_{i}$. It is commutative if for all $i$, $j$, and $k$, $p_{ij}^{k}=p_{ji}^{k}$.

Let $\left(X,(R_{i})\right)$ be a commutative association scheme with $n$ classes and $A_{i}$ be the adjacency matrix of the diagraph $(X,R_{i})$. The matrices $A_{0},A_{1},\ldots,A_{n}$ span a vector space over the complex numbers called the Bose-Mesner algebra of $\left(X,(R_{i})\right)$ with dimension $n+1$. This vector space has another uniquely defined basis consisting of minimal idempotent matrices $E_{0},E_{1},\ldots,E_{n}$. Then

[TABLE]

The uniquely defined numbers $P_{i}(k)$ and $Q_{k}(i)$ are called the $P$-numbers and the $Q$-numbers of $\left(X,(R_{i})\right)$, respectively.

Let $V$ be a vector space over the finite field $\mathbb{F}_{q}$ of dimension $m$ and $X(m,q)$ be the set of symmetric bilinear forms on $V$, where $q$ is an odd prime power. Let $\alpha_{1},\ldots,\alpha_{m}$ be a basis of $V$. Then a symmetric bilinear form $B\in X(m,q)$ has the $m\times m$ symmetric matrix

[TABLE]

The rank of $B$ is the rank of this matrix, which is independent of the choice of the basis. This matrix is congruent to a diagonal matrix, whose diagonal is either zero of $[z,1,\ldots,1,0,\ldots,0]$ for some nonzero $z\in\mathbb{F}_{q}$. The type of $B$ is $\eta(z)$, where $\eta$ is the quadratic character of $\mathbb{F}_{q}$.

Let $X_{r,\tau}$ be the set of all the symmetric bilinear forms with rank $r$ and type $\tau$. Define

[TABLE]

Then $\left(X(m,q),(R_{r,\tau})\right)$ is an association scheme with $2m$ classes.

Let $Y$ be a subset of $X(m,q)$ and the inner distribution of $Y$ be the sequence of numbers $(a_{r,\tau})$, where

[TABLE]

Let $Q_{k,\epsilon}(r,\tau)$ be the $Q$-numbers of $\left(X(m,q),(R_{r,\tau})\right)$. The dual inner distribution of $Y$ is the sequence of $(a_{k,\epsilon}^{\prime})$, where

[TABLE]

Note that $\tau\in\{1,-1\}$. When $r=0$, we write $a_{0}$.

Definition 1

Let $Y$ be a subset of $X(m,q)$. The set $Y$ is a $d$-code if $a_{i,1}=a_{i,-1}=0$ for each $i\in\{1,2,\ldots,d-1\}$. The set $Y$ is a proper $d$-code if it is a $d$-code and it is not a $(d+1)$-code. The set $Y$ is a $t$-design if $a_{i,1}^{\prime}=a_{i,-1}^{\prime}=0$ for each $i\in\{1,2,\ldots,t\}$. The set $Y$ is a $(2t+1,\epsilon)$-design if it is a $(2t+1)$-design and $a_{2t+2,\epsilon}^{\prime}=0$.

Note that the designs involved in associate schemes are not the usual t-designs studied in combinatorial design theory. Define $q^{2}$-analogs of binomial coefficients

[TABLE]

for integers $n$ and $k\geq 0$. Note that $\begin{bmatrix}n\\ 0\end{bmatrix}=1$. The inner distribution of $Y$ can be given by $q^{2}$-analogs of binomial coefficients in the following theorem and proposition.

Theorem 2.1 (Theorem 3.9, S15 )

If $Y$ is a $(2\delta-1)$-code and a $(2n-2\delta+3)$-design in $X(2n+1,q)$, then the inner distribution $(a_{r,\tau})$ of $Y$ satisfies

[TABLE]

for $i>0$. If $Y$ is a $(2\delta-1)$-code and a $(2n-2\delta+2)$-design in $X(2n,q)$, then the inner distribution $(a_{r,\tau})$ of $Y$ satisfies

[TABLE]

for $i>0$.

Proposition 2 (Proposition 3.10, S15 )

If $Y$ is a $(2\delta)$-code and a $(2n-2\delta+1)$-design in $X(2n,q)$, then the inner distribution $(a_{r,\tau})$ of $Y$ satisfies

[TABLE]

for $i>0$. If $Y$ is a $(2\delta)$-code and a $(2n-2\delta+1,\eta(-1)^{n-\delta+1})$-design in $X(2n+1,q)$, then the inner distribution $(a_{r,\tau})$ of $Y$ satisfies

[TABLE]

for $i>0$.

3 A class of linear codes of

length $n=\frac{q^{m}-1}{2}$

In this section, let $q$ be an odd prime power, $m$ be a positive integer, $\alpha$ be a generator of $\mathbb{F}_{q^{m}}^{*}$ and $\beta=\alpha^{2}$. Then $\beta$ is a primitive $n$-th root of unity in $\mathbb{F}_{q^{m}}$, where $n=\frac{q^{m}-1}{2}$.

Let $h$ be a positive integer. When $m$ is odd, define the codes

[TABLE]

and

[TABLE]

When $m$ is even, define the codes

[TABLE]

and

[TABLE]

Let $Q(x)$ be a quadratic form defined by

[TABLE]

where $k_{j}\geq 0$ and $a_{j}\in\mathbb{F}_{q^{m}}$. The symmetric bilinear form associated with $Q(x)$ is

[TABLE]

Let $h\geq 1$. Define the set of quadratic forms:

[TABLE]

where $m$ is odd. Define the set of quadratic forms:

[TABLE]

where $m$ is even. Then we have the following two sets of bilinear forms associated with $Q_{1}$ and $Q_{2}$ respectively.

[TABLE]

and

[TABLE]

Then $|S_{1}|=|S_{2}|=q^{m(\frac{m+1}{2}-h)}$. The inner distributions of $S_{1}$ and $S_{2}$ are given in the following proposition.

Proposition 3

Let $(a_{0},a_{1,1},a_{1,-1},\ldots,a_{m,1},a_{m,-1})$ be the inner distribution of $S_{i}$. If $m$ is odd, then $S_{1}$ is a proper $(2h+1)$-code and $(m+1-2h)$-design in $X(m,q)$, and the inner distribution of $S_{1}$ is given in Theorem 2.1. If $m$ is even, then $S_{2}$ is a proper $(2h)$-code and $(m+1-2h)$-design in $X(m,q)$, and the inner distribution of $S_{2}$ is given in Proposition 2.

Proof

We first prove that $S_{1}$ is a $(m+1-2h)$-design in $X(m,q)$, where $m$ is odd.

Let $U$ be a $t$-dimensional subspace of $\mathbb{F}_{q^{m}}$ and $A$ be a symmetric bilinear form on $U$, where $t=m+1-2h$. For $a=(a_{0},a_{1},\ldots,m_{t-1})\in\mathbb{F}_{q^{m}}^{t}$, define a bilinear form $B_{a}$ on $\mathbb{F}_{q^{m}}^{t}$

[TABLE]

From Lemma 4.6 in S15 , the set $\{B_{a}\mid_{U}:a\in\mathbb{F}_{q^{m}}^{t}\}$ is a multiset in which each bilinear form on $U$ occurs a constant number (depending only on $t$) of times. From Lemma 4.3 in L17 , the number of elements in the multiset $\left\{D+D^{\prime}:D\in\{B_{a}:a\in\mathbb{F}_{q^{m}}^{t}\}\right\}$ that are an extension of $A$ is a constant independent of $U$ and $A$.

For $a=(a_{0},a_{1},\ldots,m_{t-1})\in\mathbb{F}_{q^{m}}^{t}$, we have

[TABLE]

Then

[TABLE]

Note that $\{a_{j-h},a_{m-j-h}:h\leq j\leq\frac{m-1}{2}\}=\{a_{j}:0\leq j\leq t-1\}$. Then $a_{j-h}+a_{m-j-h}^{sj}$ ranges over $\mathbb{F}_{q^{m}}$ for $q^{m}$ times when $a_{j-h}$ and $a_{m-j-h}$ ranges over $\mathbb{F}_{q^{m}}$. Hence, the number of elements in $S_{1}$ which are an extension of $A$ is a constant independent of $U$ and $A$. From Theorem 3.11 in S15 , the set $S_{1}$ is a $t$-design in $X(m,q)$.

We then prove that $S_{1}$ is a proper $(2h+1)$-code. For a bilinear form

[TABLE]

we have

[TABLE]

Since the degree of the linearized polynomial $\sum_{j=h}^{\frac{m-1}{2}}\left(a_{j}^{-h}x^{q^{j-h}}+a_{j}^{q^{m-j-h}}x^{q^{m-j-h}}\right)y$ over $\mathbb{F}_{q^{m}}$ is at most $m-2h$, the dimension of the vector space $Rad(B(x,y))$ is at most $m-2h$. Hence, $rank(B(x,y))=m-dim(Rad(B(x,y)))\geq 2h$. Hence, $S_{1}$ is a $(2h)$-code in $X(m,q)$. Since $S_{1}$ is a $(2m+1-2h)$-design, then $S_{1}$ is a $(2m-2h,\tau)$-design. From Proposition 2 with $\delta=h$, we have $a_{2h,\tau}=0$ for $\tau=\{1,-1\}$. Hence, $S_{1}$ is a $(2h+1)$-code. From Theorem 2.1, we have $a_{2h+1,\tau}>0$. Hence, $S_{1}$ is a proper $(2h+1)$-code.

From a similar discussion, we have the corresponding results for $S_{2}$ when $m$ is even.

Lemma 4

Let $q$ be odd and $Q$ be a quadratic form of rank $r\geq 1$ and type $\tau$ on $\mathbb{F}_{q^{m}}$. Then the weight enumerator $W_{r,\tau}$ of $\left(Q(\alpha^{l})+a\right)_{l=0}^{n-1}$ is

[TABLE]

if $r$ is odd, or

[TABLE]

if $r$ is even.

Proof

Let $f(x)=Q(x)+a$. We just computer the weight $wt(f)$. Let $N(f=0)$ be the number of solutions of $f(x)=0$.

When $a=0$, from Lemma 3, we have

[TABLE]

Then

[TABLE]

When $a\neq 0$ and $\eta(-a)=1$, from Lemma 3, we have

[TABLE]

Then

[TABLE]

When $a\neq 0$ and $\eta(-a)=-1$, from Lemma 3, we have

[TABLE]

Then

[TABLE]

Hence, this lemma follows.

Lemma 5

Let $q$ be odd and $Q$ be a quadratic form of rank $r\geq 1$ and type $\tau$ on $\mathbb{F}_{q^{m}}$. Then the weight enumerator $W_{r,\tau}$ of $\left(Q(\alpha^{l})\right)_{l=0}^{n-1}$ is $Z^{\frac{q^{m}-q^{m-1}}{2}}$ (resp. $Z^{\frac{q^{m}-q^{m-1}-\tau\eta(-1)^{\frac{r}{2}}(q-1)q^{m-\frac{r+2}{2}}}{2}})$ if $r$ is odd (resp. even).

Proof

From results of $a=0$ in Lemma 4, this lemma follows.

We determine the weight enumerators of $\mathcal{C}_{1,h}$, $\tilde{\mathcal{C}}_{1,h}$, $\mathcal{C}_{1,h}$, and $\tilde{\mathcal{C}}_{1,h}$ in the following theorem.

Theorem 3.1

When $m$ is odd, the weight enumerator of the code $\mathcal{C}_{1,h}$ is

[TABLE]

where $a_{r,\tau}$ is given in Theorem 2.1 and $W_{r,\tau}$ is given in Lemma 4. The weight enumerator of the code $\tilde{\mathcal{C}}_{1,h}$ is

[TABLE]

where $a_{r,\tau}$ is given in Theorem 2.1 and $W_{r,\tau}$ is given in Lemma 5. When $m$ is even, the weight enumerator of the code $\mathcal{C}_{2,h}$ is

[TABLE]

where $a_{r,\tau}$ is given in Proposition 2 and $W_{r,\tau}$ is given in Lemma 4. The weight enumerator of the code $\tilde{\mathcal{C}}_{2,h}$ is

[TABLE]

where $a_{r,\tau}$ is given in Proposition 2 and $W_{r,\tau}$ is given in Lemma 5.

Proof

We first give the weight enumerator of $\mathcal{C}_{1,h}$. When $m$ is odd, we have

[TABLE]

The weight enumerator of $\{(a)_{l=0}^{n-1}:a\in\mathbb{F}_{q}\}$ is $1+(q-1)\frac{q^{m}-1}{2}$. From Proposition 3, $S_{1}$ is a proper $(2h+1)$-code. For any nonzero $Q(x)\in Q_{1}$, we have $rank(Q(x))=r\geq 2h+1$. The weight enumerator $W_{r,\tau}$ of $\left(Q(\alpha^{l})+a\right)_{l=0}^{n-1}$ with rank $r>0$ is given in Lemma 4. The inner distribution of $Q_{1}$ is just the inner distribution of $S_{1}$, which is given in Theorem 2.1. Hence, we have the weight enumerator of $\mathcal{C}_{1,h}$

[TABLE]

From the similar method, we have the other results. Hence, this theorem follows.

Lemma 6

When $m\geq 2$ and $1\leq i\leq\lfloor\frac{m+3}{4}\rfloor$, then $\delta_{i}=\frac{q^{m}-q^{m-1}}{2}-1-\frac{q^{\lfloor\frac{m-3}{2}\rfloor+i}-1}{2}$ is the $i$-th largest coset leader module $n$.

Proof

When $q=3$, from LDXG17 , this lemma holds. Suppose that $q>3$. When $m$ is odd, then the $q$-adic expansion of $\delta_{i}$ is

[TABLE]

and its cyclotomic coset $C_{\delta_{i}}$ is

[TABLE]

From Lemma 9 in ZSK19 , $\delta_{1}$ is the largest coset leader module $n$. Then we just prove that for $i>1$, there does not exist a coset leader $s$ satisfying $\delta_{i}<s<\delta_{i-1}$.

Suppose such a coset leader $s$ satisfying $\delta_{i}<s<\delta_{i-1}$ exists. From Lemma 8 in ZSK19 , then the $q$-adic expansion of $s$ is

[TABLE]

where $k=\frac{m-3}{2}+i-1$, $\frac{q-3}{2}\leq s_{j}\leq q-1$ and $(\frac{q-1}{2})\sum_{j=0}^{k}q^{j}<\sum_{j=0}^{k}s_{j}q^{j}<(q-1)q^{k}+(\frac{q-1}{2})\sum_{j=0}^{k-1}q^{j}$.

When $s_{k}=\frac{q-1}{2}$, from $t>\delta_{i}$, we have $(q^{m-2-t}s\mod n)<s$, where $0\leq t<k$ and $t$ is the largest index such that $s_{t}>\frac{q-1}{2}$. It is a contradiction to the coset leader $s$.

When $\frac{q-1}{2}<s_{k}<q-1$, then $(q^{m-1-k}s\mod n)<s$, which is a contradiction to the coset leader $s$.

When $s_{k}=q-1$, from $s<\delta_{i-1}$, there exists a $t$ satisfying $(q^{m-1-t}s\mod n)<s$ or $(q^{m-2-t}s\mod n)<s$, where $0\leq t<k$. It is a contradiction to the coset leader $s$.

Hence, $\delta_{i}$ is the $i$-th largest coset leader module $n$ when $m$ is odd. From the similar method, it holds when $m$ is even.

The following theorem gives parameters of the BCH code $\mathcal{C}_{(n,q,m,\delta_{i})}$.

Theorem 3.2

Let $m\geq 2$, $1\leq i\leq\lfloor\frac{m+3}{4}\rfloor$, and $h=\lfloor\frac{m}{2}\rfloor-i+1$. Then the BCH code $\mathcal{C}_{(n,q,m,\delta_{i})}$ is $\mathcal{C}_{1,h}$ (resp. $\mathcal{C}_{2,h}$) for $m$ odd (resp. even) and it has parameters $[n,(i+1+\frac{m-1}{2}-\lfloor\frac{m}{2}\rfloor)m,\delta_{i}]$ and the Bose distance $d_{B}=\delta_{i}$. The code $\tilde{\mathcal{C}}_{(n,q,m,\delta_{i})}$ is $\tilde{\mathcal{C}}_{1,h}$ (resp. $\tilde{\mathcal{C}}_{2,h}$) for $m$ odd (resp. even) and it has dimension $(i+\frac{m-1}{2}-\lfloor\frac{m}{2}\rfloor)m$.

Proof

Note that $\delta_{i}$ is the $i$-th largest coset leader module $n$. When $m$ is odd, we have $l_{i}=\mid C_{\delta_{i}}\mid=m$ and

[TABLE]

Hence the code $\mathcal{C}_{(n,q,m,\delta_{i})}=\mathcal{C}_{1,h}$ and the dimension of $\mathcal{C}_{(n,q,m,\delta_{i})}$ is $(i+1+\frac{m-1}{2}-\lfloor\frac{m}{2}\rfloor)m$. When $r=2h+1$, the $\left(Q(\alpha^{l})+a\right)_{l=0}^{n-1}$ has a codeword of weight $\delta_{i}$. From Lemma 1, the code $\mathcal{C}_{(n,q,m,\delta_{i})}$ has Bose distance $\delta_{i}$. Hence, the code $\mathcal{C}_{(n,q,m,\delta_{i})}$ has parameters $[n,(i+1+\frac{m-1}{2}-\lfloor\frac{m}{2}\rfloor)m,\delta_{i}]$ and Bose distance $\delta_{i}$. From the similar discussion, we have the other results of this proposition.

Example 1

Let $q=3$, $h=\lfloor\frac{m}{2}\rfloor$, and $i=1$. When $m$ is odd (resp. even), the code $\mathcal{C}_{(n,3,m,\delta_{1})}$ is $\mathcal{C}_{1,\frac{m-1}{2}}$ ( resp. $\mathcal{C}_{2,\frac{m}{2}}$), and it has parameters $[\frac{3^{m}-1}{2},m+1,\delta_{1}]$ (resp. $[\frac{3^{m}-1}{2},\frac{m+2}{2},\delta_{1}]$ ) and the weight distribution in Table II (resp. Table I) in Theorem 19 LDXG17 . When $m$ is odd (resp. even), the code $\tilde{\mathcal{C}}_{(n,3,m,\delta_{1})}$ is $\tilde{\mathcal{C}}_{1,\frac{m-1}{2}}$ ( resp. $\tilde{\mathcal{C}}_{2,\frac{m}{2}}$), and it has parameters $[\frac{3^{m}-1}{2},m,3^{m-1}]$ (resp. $[\frac{3^{m}-1}{2},\frac{m}{2},3^{m-1}+3^{\frac{m-2}{2}}]$ ) and the weight enumerator $1+(3^{m}-1)Z^{3^{m-1}}$ (resp. $1+(3^{\frac{m}{2}}-1)Z^{3^{m-1}+3^{\frac{m-2}{2}}}$) in Theorem 22 LDXG17 .

Example 2

Let $q=3$, $h=\lfloor\frac{m}{2}\rfloor-1$, and $i=2$. When $m$ is odd (resp. even), the code $\mathcal{C}_{(n,3,m,\delta_{2})}$ is $\mathcal{C}_{1,\frac{m-3}{2}}$ ( resp. $\mathcal{C}_{2,\frac{m-2}{2}}$), and it has parameters $[\frac{3^{m}-1}{2},2m+1,\delta_{2}]$ (resp. $[\frac{3^{m}-1}{2},\frac{3m+2}{2},\delta_{2}]$ ) and the weight distribution in Table VI (resp. Table V) in Theorem 29 LDXG17 . When $m$ is odd (resp. even), the code $\tilde{\mathcal{C}}_{(n,3,m,\delta_{2})}$ is $\tilde{\mathcal{C}}_{1,\frac{m-3}{2}}$ ( resp. $\tilde{\mathcal{C}}_{2,\frac{m-2}{2}}$), and it has parameters $[\frac{3^{m}-1}{2},2m,3^{m-1}-3^{\frac{m-1}{2}}]$ (resp. $[\frac{3^{m}-1}{2},\frac{3m}{2},3^{m-1}-3^{\frac{m-2}{2}}]$ ) and the weight distribution in Table IV (resp. Table III) in Theorem 26 LDXG17 .

Example 3

When $h=\lfloor\frac{m}{2}\rfloor$ and $i=1$, the code $\mathcal{C}_{(n,q,m,\delta_{1})}$ (resp. $\tilde{\mathcal{C}}_{(n,q,m,\delta_{1})}$ ) is given in Theorem 5 (resp. Theorem 4) in ZSK19 . When $h=\lfloor\frac{m}{2}\rfloor-1$ and $i=2$, the code $\mathcal{C}_{(n,q,m,\delta_{2})}$ (resp. $\tilde{\mathcal{C}}_{(n,q,m,\delta_{2})}$ ) is given in Theorem 7 (resp. Theorem 6) in ZSK19 .

Remark 1

Let $q=3$, $h=\lfloor\frac{m}{2}\rfloor-1$, and $i=2$. The code $\tilde{\mathcal{C}}_{(n,3,m,\delta_{2})}$ is a three-weight code with the weight distribution in Table 1 of TDX19 . Then these codes can be used to construct $2$-designs.

4 Conclusion

This paper studies a class of linear codes of length $\frac{q^{m}-1}{2}$ over $\mathbb{F}_{q}$ with special trace representation, uses association schemes from bilinear forms associated with quadratic forms, and determines the weight enumerators of these codes. From determining some cyclotomic coset leaders $\delta_{i}$ of cyclotomic cosets modulo $\frac{q^{m}-1}{2}$, we prove that narrow-sense BCH codes of length $\frac{q^{m}-1}{2}$ with designed distance $\delta_{i}=\frac{q^{m}-q^{m-1}}{2}-1-\frac{q^{\lfloor\frac{m-3}{2}\rfloor+i}-1}{2}$ have the corresponding trace representation and have the minimal distance $d=\delta_{i}$ and the Bose distance $d_{B}=\delta_{i}$, where $1\leq i\leq\lfloor\frac{m+3}{4}\rfloor$. It is interesting to determine parameters of more BCH codes.

Acknowledgement. This work was supported by SECODE project and the National Natural Science Foundation of China (Grant No. 11871058, 11531002, 11701129). C. Tang also acknowledges support from 14E013, CXTD2014-4 and the Meritocracy Research Funds of China West Normal University. Y. Qi also acknowledges support from Zhejiang provincial Natural Science Foundation of China (LQ17A010008, LQ16A010005).