New Construction of Maximum Distance Separable (MDS) Self-Dual Codes over Finite Fields

Abstract

Maximum distance separable (MDS) self-dual codes have useful properties due to their optimality with respect to the Singleton bound and its self-duality. MDS self-dual codes are completely determined by the length $n,$ so the problem of constructing q-ary MDS self-dual codes with various lengths is a very interesting topic. Recently X. Fang et al. using a method given in previous research, where several classes of new MDS self-dual codes were constructed through (extended) generalized Reed-Solomon codes, in this paper, based on the method given in we achieve several classes of MDS self-dual codes.

1. Introduction

Let ${\mathbb{F}}_q$ be the finite field with q elements. A q-ary $[n,k,d]$ linear code $\mathcal{C}$ is a k-dimensional subspace of ${\mathbb{F}}_q^n$ with minimum (Hamming) distance $d.$ If the parameters $[n,k,d]$ satisfy $k+d=n+1,$ the code is called an MDS (maximum distance separable) code. A self-dual code is a linear code satisfying $\mathcal{C}={\mathcal{C}}^{\perp }.$ A linear complementary-dual code is a linear code satisfying $\mathcal{C}\cap {\mathcal{C}}^{\perp }=\left\{\mathbf{0}\right\}.$

The study of MDS self-dual codes has attracted a great deal of attention in recent years due to its theoretical and practical importance. The center of the study of MDS codes includes the existence of MDS codes [1], classification of MDS codes [2], balanced MDS codes [3], non-Reed-Solomon MDS codes [4], complementary-dual MDS codes [5,6], and lowest density MDS codes [7].

As the parameters of an MDS self-dual code are completely determined by the code’s length n, the main interest here is to determine the existence and give the construction of q-ary MDS self-dual codes for various lengths. The problem is completely solved for the case where q is even [8]. Many MDS self-dual codes over finite fields of odd characteristics were constructed [9,10,11,12,13,14].

In [11], Jin and Xing constructed several classes of MDS self-dual code from generalized Reed-Solomon code. Yan generalized Jin and Xing’s method and constructed several classes of MDS self-dual codes via generalized Reed-Solomon codes and extended generalized Reed-Solomon codes [14]. In [12], Ladad, Liu and Luo produced more classes of MDS self-dual codes based on [11] and [14]. In [9], based on the [11,12,14] more new parameter MDS self-dual codes were presented. Based on the method raised in [9], we present some classes of MDS self-dual codes.

2. Preliminaries

In this section we introduce some basic notations of generalized Reed-Solomon codes and extended generalized Reed-Solomon codes. For more details, the reader is referred to [15].

Throughout this paper, q is a prime power, ${\mathbb{F}}_q$ is the finite fields with q elements and let n be a positive integer with $1<n\le q.$ For any $x\in {\mathbb{F}}_{q^2},$ we denote by $\overline{x}$ the conjugation of $x.$ Given an $[n,k,d]$ linear code $\mathcal{C}$, its Euclidean dual code (resp. Hermitian dual code) is denoted by ${\mathcal{C}}^{\perp }$ (resp. ${\mathcal{C}}^{{\perp }_H}$). The codes ${\mathcal{C}}^{\perp }$ and ${\mathcal{C}}^{{\perp }_H}$ are defined by

$${\mathcal{C}}^{\perp }=\{x=(x_1,x_2,\dots ,x_n)\in {\mathbb{F}}_q^n:\sum _{i=1}^n{x}_i{y}_i=0,\forall y=(y_1,y_2,\dots ,y_n)\in \mathcal{C}\},$$

$${\mathcal{C}}^{{\perp }_H}=\{x=(x_1,x_2,\dots ,x_n)\in {\mathbb{F}}_{q^2}^n:\sum _{i=1}^n{x}_i\overline{y_i}=0,\forall y=(y_1,y_2,\dots ,y_n)\in \mathcal{C}\},$$

respectively. In this paper, we only consider the Euclidean inner product.

Let $\overrightarrow{a}=({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n),$ where ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n$ are n distinct elements of ${\mathbb{F}}_q$. Fix n nonzero elements $v_1,v_2,\dots ,v_n$ of ${\mathbb{F}}_q$ ($v_i$ are not necessarily distinct), put $\overrightarrow{v}=(v_1,v_2,\dots ,v_n).$ For $1\le k\le n,$ the k-dimensional generalized Reed-Solomon code (GRS for short) of length n associated with $\overrightarrow{a}$ and $\overrightarrow{v}$ is defined to be

$${\mathbf{GRS}}_k(\overrightarrow{a},\overrightarrow{v})=\{(v_{1}f\left({\alpha }_1\right),v_{2}f\left({\alpha }_2\right),\dots ,v_{n}f\left({\alpha }_n\right)):f\left(x\right)\in {\mathbb{F}}_q\left[x\right],deg\left(f\left(x\right)\right)\le k-1\}.$$

(1)

It is well known that the code ${\mathbf{GRS}}_k(\overrightarrow{a},\overrightarrow{v})$ is a q-ary $[n,k,n-k+1]$ MDS code and the dual of a GRS code is again a GRS MDS code; indeed

$${\mathbf{GRS}}_k^{\perp }(\overrightarrow{a},\overrightarrow{v})={\mathbf{GRS}}_{n-k}(\overrightarrow{a},{\overrightarrow{v}}^{\prime })$$

for some ${\overrightarrow{v}}^{\prime }=(v_1^{\prime },v_2^{\prime },\dots ,v_n^{\prime })$ with $v_i^{\prime }\ne 0$ for all $1\le i\le n$ (e.g., see [15]).

Furthermore, the extended generalized Reed-Solomon code ${\mathbf{GRS}}_k(\overrightarrow{a},\overrightarrow{v},\infty )$ given by

$${\mathbf{GRS}}_k(\overrightarrow{a},\overrightarrow{v},\infty )=\{(v_{1}f\left({\alpha }_1\right),v_{2}f\left({\alpha }_2\right),\dots ,v_{n}f\left({\alpha }_n\right),f_{k-1}):f\left(x\right)\in {\mathbb{F}}_q\left[x\right],deg\left(f\left(x\right)\right)\le k-1\},$$

(2)

where $f_{k-1}$ stands for the coefficient of $x^{k-1}$ in $f\left(x\right).$ It is also well known that ${\mathbf{GRS}}_k(\overrightarrow{a},\overrightarrow{v},\infty )$ is a q-ary $[n+1,k,n-k+2]$ MDS code and the dual code is also a GRS MDS code (e.g., see [15]).

Put $\overrightarrow{a}=({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n)$ and denote by ${\mathcal{A}}_{\overrightarrow{a}}$ the matrix

$$\begin{array}{c}\left(\begin{array}{cccc}1& 1& \dots & 1\\ {\alpha }_1& {\alpha }_2& \dots & {\alpha }_n\\ {\alpha }_1^2& {\alpha }_2^2& \dots & {\alpha }_n^2\\ ⋮& ⋮& \ddots & ⋮\\ {\alpha }_1^{n-2}& {\alpha }_2^{n-2}& \dots & {\alpha }_n^{n-2}\end{array}\right)\end{array}$$

Lemma 1

([11]). The solution space of the equation system ${\mathcal{A}}_{\overrightarrow{a}}{X}^T=\mathbf{0}$ has dimension 1 and $\{\overrightarrow{u}=(u_1,u_2,\dots ,u_n)\}$ is a basis of this solution space, where $u_i={\prod }_{1\le j\le n,j\ne i}{({\alpha }_i-{\alpha }_j)}^{-1}.$ Furthermore, for any two polynomials $f\left(x\right),g\left(x\right)\in {\mathbb{F}}_q\left[x\right]$ with $deg\left(f\right)\le k-1$ and $deg\left(g\right)\le n-k-1$, one has ${\sum }_{i=1}^{n}f\left({\alpha }_i\right)\left(u_{i}g\left({\alpha }_i\right)\right)=0.$

We define

$$L_{\overrightarrow{a}}\left({\alpha }_i\right)=\prod _{1\le j\le n,j\ne i}({\alpha }_i-{\alpha }_j).$$

The conclusion of the following lemma is straightforward. For completeness, we provide its proof.

Lemma 2

([11]). Let n be an even number, if there exists $\lambda \in {\mathbb{F}}_q^{*}$ such that $\lambda L_{\overrightarrow{a}}\left({\alpha }_i\right)$ is square element for all $i=1,2,\dots ,n,$ then the code ${\mathbf{GRS}}_{n/2}(\overrightarrow{a},\overrightarrow{v})$ defined in (1) is MDS self-dual code of length n.

Proof. 

Let $f\left(x\right),g\left(x\right)\in {\mathbb{F}}_q\left[x\right]$ with $deg\left(f\right)\le \frac{n}{2}-1$ and $deg\left(g\right)\le \frac{n}{2}-1.$ By Lemma 1, we have ${\sum }_{i=1}^{n}f\left({\alpha }_i\right)\left(u_{i}g\left({\alpha }_i\right)\right)=0,$ where $u_i={\prod }_{1\le j\le n,j\ne i}{({\alpha }_i-{\alpha }_j)}^{-1}$ for $i=1,2,\dots ,n.$ Hence,

$$0=\lambda \sum _{i=1}^{n}f\left({\alpha }_i\right)\left(u_{i}g\left({\alpha }_i\right)\right)=\sum _{i=1}^{n}f\left({\alpha }_i\right)\left(\lambda u_{i}g\left({\alpha }_i\right)\right)=\sum _{i=1}^n\left(v_{i}f\left({\alpha }_i\right)\right)\left(v_{i}g\left({\alpha }_i\right)\right)(\mathrm{since}\lambda u_i=v_i^2).$$

This implies that ${\mathbf{GRS}}_{n/2}^{\perp }(\overrightarrow{a},\overrightarrow{v})={\mathbf{GRS}}_{n/2}(\overrightarrow{a},\overrightarrow{v}).$ □

H. Yan [14] observed the following two results.

Lemma 3

([14]). Let n be an even integer and $k=\frac{n}{2}.$ If $-L_{\overrightarrow{a}}\left({\alpha }_i\right)$ is square element for all $i=1,2,\dots ,n-1,$ then the code ${\mathbf{GRS}}_k(\overrightarrow{a},\overrightarrow{v},\infty )$ defined in (2) is MDS self-dual code of length $n.$

Lemma 4

([14]). Let $m\mid q-1$ be a positive integer and let $\alpha \in {\mathbb{F}}_q$ be a primitive m-th root of unity. Then for any $1\le i\le m,$ we have

$$\prod _{1\le j\le m,j\ne i}({\alpha }^i-{\alpha }^j)=m{\alpha }^{-i}.$$

3. Main Result

Let $q=r^2,$ where r is odd prime power, ${\mathbb{F}}_q$ be the finite fields with q elements. Suppose $m\mid q-1,\alpha$ is a primitive m-th root of unity and $\mathbf{H}=<\beta >$ is the cyclic group generated by $\beta .$

Theorem 1.

Let $q=r^2,$ where r is an odd prime power, $r\equiv 1\left(\mathrm{mod}4\right)$. Suppose that $m\mid (q-1)$ and $\frac{q-1}{m}$ is even, $m\equiv 0\left(\mathrm{mod}4\right).$ If $1\le t\le \frac{2(r+1)}{gcd\left(2\right(r+1),m)}.$ Then there exists an $[n=tm,\frac{n}{2}]$-MDS self-dual code.

Proof. 

Let $\alpha$ be a primitive m-th root of unity and $\mathbf{H}=<\beta >$ is the cyclic group of order $2(r+1).$ By the theorem of group homomorphism,

$$(\mathbf{H}\times \langle \alpha \rangle )/\langle \alpha \rangle \cong \mathbf{H}/(\mathbf{H}\cap \langle \alpha \rangle ).$$

Let $i_1,i_2,\dots ,i_t$ be t distinct elements, such that $0\le i_1<i_2<\cdots <i_t<2(r+1).$ Denote $I=\{i_1,i_2,\dots ,i_t\},A=i_1+i_2+\cdots +i_t$ and $\mathbf{B}=\{{\beta }^{i_1},{\beta }^{i_2},\dots ,{\beta }^{i_t}\}$ be a set of coset representatives of $(\mathbf{H}\times \langle \alpha \rangle )/\langle \alpha \rangle$. Let

$$\overrightarrow{a}=(\alpha {\beta }^{i_1},\dots ,{\alpha }^m{\beta }^{i_1},\alpha {\beta }^{i_2},\dots ,{\alpha }^m{\beta }^{i_2},\dots ,\alpha {\beta }^{i_t},\dots ,{\alpha }^m{\beta }^{i_t}).$$

Then the entries of $\overrightarrow{a}$ are distinct in ${\mathbb{F}}_q^{*}.$

It is known that $x^m-y^m={\displaystyle \prod _{j=1}^m}(x-{\alpha }^{j}y)$. By the statement of Lemma 3, we get

$$\begin{array}{lll}{L}_{\overrightarrow{a}}\left({\beta }^z{\alpha }^k\right)& =& \prod _{1\le j\le m,j\ne k}({\beta }^z{\alpha }^k-{\beta }^z{\alpha }^j)\prod _{l\in I,l\ne z}\prod _{j=1}^m({\beta }^z{\alpha }^k-{\beta }^l{\alpha }^j)\\ & =& {\beta }^{z(m-1)}\prod _{1\le j\le m,j\ne k}({\alpha }^k-{\alpha }^j)\prod _{l\in I,l\ne z}[{\left({\beta }^z{\alpha }^k\right)}^m-{\beta }^{lm}]\hfill \\ & =& {\beta }^{z(m-1)}m{\alpha }^{-k}\prod _{l\in I,l\ne z}({\beta }^{zm}-{\beta }^{lm}).\hfill \end{array}$$

Let $v={\displaystyle \prod _{l\in I,l\ne z}}({\beta }^{zm}-{\beta }^{lm}),$ then

$$\begin{array}{lll}{v}^r& =& \prod _{l\in I,l\ne z}({\beta }^{zmr}-{\beta }^{lmr})(\mathrm{since}{\beta }^{2(r+1)}=1,{\beta }^r=-{\beta }^{-1})\\ & =& \prod _{l\in I,l\ne z}[{(-{\beta }^{-1})}^{zm}-{(-{\beta }^{-1})}^{lm}]\hfill \\ & =& \prod _{l\in I,l\ne z}[{\left({\beta }^{-1}\right)}^{zm}-{\left({\beta }^{-1}\right)}^{lm}]\hfill \\ & =& \prod _{l\in I,l\ne z}{\left({\beta }^{-1}\right)}^{zm+lm}({\beta }^{lm}-{\beta }^{zm})\hfill \\ & =& {(-1)}^{t-1}{\beta }^{-(A+(t-2\left)z\right)m}v\hfill \end{array}$$

So $v^{r-1}={(-1)}^{t-1}{\beta }^{-(A+(t-2\left)z\right)m}.$

Let g be a generator of ${\mathbb{F}}_q^{*}$, then $\alpha =g^{\frac{q-1}{m}},\beta =g^{\frac{r-1}{2}},-1=g^{\frac{r^2-1}{2}},v=g^{\frac{r+1}{2}(t-1)-(A+(t-2)z)\frac{m}{2}+i(r+1)}.$ Note that $\beta ,m$ and $\alpha$ are square elements of ${\mathbb{F}}_q^{*},$ we take $\lambda =g^{\frac{r+1}{2}(t-1)},$ then $\lambda L_{\overrightarrow{a}}\left({\beta }^z{\alpha }^k\right)$ is a square element of ${\mathbb{F}}_q^{*}$.

This implies there exists a q-ary $[n,\frac{n}{2}]$ MDS self-dual code. □

Example 1.

Let $r=173,q={173}^2,r\equiv 1\left(\mathrm{mod}4\right),m=4\times 43,\frac{q-1}{m}=174$ is even. For $1\le t\le \frac{2(r+1)}{gcd\left(2\right(r+1),m)}=87,$ we choose $t=81.$ By Theorem 1, there exists the MDS self-dual code with length $n=mt=$ 13,932.

Theorem 2.

Let $q=r^2,$ where r is an odd prime power. Suppose that m is odd, $m\mid (q-1)$ and $\frac{q-1}{m}$ is even. If $1\le t\le min\{\frac{r+1}{gcd\left(2\right(r+1),m)},\frac{r+1}{2}\}$ and t is odd, then there exists a q-ary $[n=tm+1,\frac{n}{2}]$ MDS self-dual code over ${\mathbb{F}}_q.$

Proof. 

Let $\alpha$ and $\beta$ be the same as in Theorem 1, we choose t distinct even number $i_1,i_2,\dots ,i_t$, $0\le i_1<i_2<\cdots <i_t<2(r+1).$ Denote $I=\{i_1,i_2,\cdots ,i_t\},A=i_1+i_2+\dots +i_t$. Suppose all $i_j\equiv 2\left(\mathrm{mod}4\right),j=1,2,\cdots ,t.$ The proof is as similar as in Theorem 1. We get

$$L_{\overrightarrow{a}}\left({\beta }^z{\alpha }^k\right)={\beta }^{z(m-1)}m{\alpha }^{-k}\prod _{l\in I,l\ne z}({\beta }^{zm}-{\beta }^{lm}).$$

Let $v={\displaystyle \prod _{l\in I,l\ne z}}({\beta }^{zm}-{\beta }^{lm}),$ then we get

$$v^{r-1}={(-1)}^{t-1}{\beta }^{-(A+(t-2\left)z\right)m},v=g^{\frac{r+1}{2}(t-1)-\frac{(A+(t-2\left)z\right)m}{2}+i(r+1)},$$

since $\frac{A+(t-2)z}{2}$ is even, it implies that v is a square element of ${\mathbb{F}}_q^{*}.$ So $-L_{\overrightarrow{a}}\left({\beta }^z{\alpha }^k\right)$ is square element of ${\mathbb{F}}_q^{*}$. By Lemma 3, there exists a q-ary $[n,\frac{n}{2}]$ MDS self-dual code. □

Example 2.

Let $r=67,q={67}^2,m=11,\frac{q-1}{m}=408$ is even. Since $2(r+1)=136=4\times 34,$ for $1\le t\le \frac{r+1}{gcd\left(2\right(r+1),m)}=68,$ we choose $t=27.$ By Theorem 2, there exists the MDS self-dual code with length $n=mt+1=298.$

Theorem 3.

Let $q=r^2,$ where r is an odd prime power, $r\equiv 1\left(\mathrm{mod}4\right)$. Suppose that m is odd, $m\mid (q-1)$ and $\frac{q-1}{m}$ is even. If $1\le t\le min\{\frac{r+1}{gcd\left(2\right(r+1),m)},\frac{r+1}{2}\}$ and t is odd, then there exists a q-ary $[n=tm+1,\frac{n}{2}]$ MDS self-dual code over ${\mathbb{F}}_q$.

Proof. 

Let $\alpha$ and $\beta$ be the same as in Theorem 1, we choose t distinct even number $i_1,i_2,\dots ,i_t$, $0\le i_1<i_2<\cdots <i_t<2(r+1).$ Denote $I=\{i_1,i_2,\cdots ,i_t\},A=i_1+i_2+\dots +i_t$, and $i_j\equiv 2\left(\mathrm{mod}4\right),j=1,2,\cdots ,t.$ We define the generalized Reed -Solomon code ${\mathbf{GRS}}_k(\overrightarrow{a},\overrightarrow{v})$ with

$$\overrightarrow{a}=(0,\alpha {\beta }^{i_1},\dots ,{\alpha }^m{\beta }^{i_1},\alpha {\beta }^{i_2},\dots ,{\alpha }^m{\beta }^{i_2},\dots ,\alpha {\beta }^{i_t},\dots ,{\alpha }^m{\beta }^{i_t}).$$

For any $z\in I$ and $1\le k\le m,$ we get

$$\begin{array}{ccc}\hfill L_{\overrightarrow{a}}\left({\beta }^z{\alpha }^k\right)& =& {\beta }^z{\alpha }^k\prod _{1\le j\le m,j\ne k}({\beta }^z{\alpha }^k-{\beta }^z{\alpha }^j)\prod _{l\in I,l\ne z}\prod _{j=1}^m({\beta }^z{\alpha }^k-{\beta }^l{\alpha }^j)\hfill \\ & =& {\beta }^{zm}m\prod _{l\in I,l\ne z}({\beta }^{zm}-{\beta }^{lm})\hfill \end{array}$$

and

$$L_{\overrightarrow{a}}\left(0\right)=\prod _{l\in I}\prod _{j=1}^m(0-{\beta }^l{\alpha }^j)={(-1)}^{mt}{\alpha }^{\frac{m(m+1)}{2}}{\left(\prod _{l\in I}{\beta }^l\right)}^m.$$

Since $r\equiv 1\left(\mathrm{mod}4\right),\frac{q-1}{m}$ is even, so $\alpha ,\beta ,m,-1$ are square elements of ${\mathbb{F}}_q^{*},$ we only need to consider $v={\displaystyle \prod _{l\in I,l\ne z}}({\beta }^{zm}-{\beta }^{lm}).$ As the calculation in the proof of Theorem 1, $v=g^{\frac{r+1}{2}(t-1)-\frac{(A+(t-2\left)z\right)m}{2}+i(r+1)}.$ Since all $i_j\equiv 2\left(\mathrm{mod}4\right)$ and t is odd, so $\frac{(A+(t-2\left)z\right)m}{2}$ is even. $L_{\overrightarrow{a}}\left({\beta }^z{\alpha }^k\right)$, $L_{\overrightarrow{a}}\left(0\right)$ are square elements of ${\mathbb{F}}_q^{*}.$ By Lemma 2, there exists a q-ary $[n,\frac{n}{2}]$ MDS self-dual code. □

Example 3.

Let $r=101,r\equiv 1\left(\mathrm{mod}4\right),q={101}^2,m=75,\frac{q-1}{m}=136$ is even. Since $2(r+1)=204=4\times 51$, for $1\le t\le \frac{r+1}{gcd\left(2\right(r+1),m)}=34,$ we choose $t=33.$ By Theorem 2, there exists the MDS self-dual code with length $n=mt+1=2476.$

Theorem 4.

Let $q=r^2,$ where r is an odd prime power. Suppose that $m\mid (q-1),\frac{q-1}{m}$ is even. If $1\le t\le \frac{2(r+1)}{gcd\left(2\right(r+1),m)}$ and $tm$ is even, then there exists a q-ary $[n=tm+2,\frac{n}{2}]$ MDS self-dual code over ${\mathbb{F}}_q$.

Proof. 

Let $\alpha$ and $\beta$ be the same as in Theorem 1. We define the extended generalized Reed -Solomon code ${\mathbf{GRS}}_k(\overrightarrow{a},\overrightarrow{v},\infty )$ with

$$\overrightarrow{a}=(0,\alpha {\beta }^{i_1},\cdots ,{\alpha }^m{\beta }^{i_1},\alpha {\beta }^{i_2},\cdots ,{\alpha }^m{\beta }^{i_2},\cdots ,\alpha {\beta }^{i_t},\cdots ,{\alpha }^m{\beta }^{i_t}).$$

For any $z\in I$ and $1\le k\le m,$ we get

$$\begin{array}{ccc}\hfill L_{\overrightarrow{a}}\left({\beta }^z{\alpha }^k\right)& =& {\beta }^z{\alpha }^k\prod _{1\le j\le m,j\ne k}({\beta }^z{\alpha }^k-{\beta }^z{\alpha }^j)\prod _{l\in I,l\ne z}\prod _{j=1}^m({\beta }^z{\alpha }^k-{\beta }^l{\alpha }^j)\hfill \\ & =& {\beta }^{zm}m\prod _{l\in I,l\ne z}({\beta }^{zm}-{\beta }^{lm})\hfill \end{array}$$

and

$$L_{\overrightarrow{a}}\left(0\right)=\prod _{l\in I}\prod _{j=1}^m(0-{\beta }^l{\alpha }^j)={(-1)}^{mt}{\alpha }^{\frac{m(m+1)}{2}}{\left(\prod _{l\in I}{\beta }^l\right)}^m.$$

Case 1: If m is even, t is odd.

${\beta }^{zm},m$ and $L_{\overrightarrow{a}}\left(0\right)$ are square elements of ${\mathbb{F}}_q^{*}.$ Let $v=\prod _{l\in I,l\ne z}({\beta }^{zm}-{\beta }^{lm}),$ as the calculation in Theorem 1, $v=g^{\frac{r+1}{2}(t-1)-\frac{(A+(t-2\left)z\right)m}{2}+i(r+1)}.$ So we only need to consider the parity of $\frac{(A+(t-2\left)z\right)m}{2}.$

• $i_1,i_2,\dots ,i_t$ are even number, so $A+(t-2)z\equiv 0\left(\mathrm{mod}2\right),v$ is a square element of ${\mathbb{F}}_q^{*}.$
• $i_1,i_2,\dots ,i_t$ are odd number, so $A+(t-2)z\equiv 0\left(\mathrm{mod}2\right),v$ is a square element of ${\mathbb{F}}_q^{*}.$

Case 2: If m and t are even, $r\equiv 3\left(\mathrm{mod}4\right),$ we assume A is an even integer. It follows that $\frac{r+1}{2}(t-1)-\frac{(A+(t-2\left)z\right)m}{2}$ is an even integer.

Case 3: If m is odd, t is even.

• $t\equiv 0\left(\mathrm{mod}4\right)$

  (1)

  If $r\equiv 1\left(\mathrm{mod}4\right),$ all $i_1,i_2,\dots ,i_t$ are odd, and $A\equiv 0\left(\mathrm{mod}4\right),$ then then $(r+1)(t-1)-(A+(t-2\left)z\right)m\equiv 0\left(\mathrm{mod}4\right),v$ is a square element of ${\mathbb{F}}_q^{*}.$

  (2)

  If $r\equiv 3\left(\mathrm{mod}4\right),$ all $i_1,i_2,\dots ,i_t$ are even, and $A\equiv 2\left(\mathrm{mod}4\right),$ then $(r+1)(t-1)-(A+(t-2\left)z\right)m\equiv 0\left(\mathrm{mod}4\right),v$ is a square element of ${\mathbb{F}}_q^{*}.$

• $t\equiv 2\left(\mathrm{mod}4\right).$

  (1)

  If $r\equiv 1\left(\mathrm{mod}4\right),A\equiv 2\left(\mathrm{mod}4\right),$ then $(r+1)(t-1)-(A+(t-2\left)z\right)m\equiv 0\left(\mathrm{mod}4\right),v$ is square of ${\mathbb{F}}_q^{*}.$

  (2)

  If $r\equiv 3\left(\mathrm{mod}4\right),A\equiv 0\left(\mathrm{mod}4\right),$ then $(r+1)(t-1)-(A+(t-2\left)z\right)m\equiv 0\left(\mathrm{mod}4\right),v$ is square of ${\mathbb{F}}_q^{*}.$

 □

We can extend the Theorem 1 to a more general case.

Theorem 5.

Let $q=r^2,$ where r is an odd prime power. Suppose that $m\mid (q-1),\frac{q-1}{m}$ is even, $s\mid m,s\mid r-1$ and $\frac{r-1}{s}$ is even. If $1\le t\le \frac{s(r+1)}{gcd\left(s\right(r+1),m)},$ then there exists a q-ary $[n=tm,\frac{n}{2}]$ MDS self-dual code over ${\mathbb{F}}_q.$

Proof. 

Let $\alpha$ be a primitive m-th root of unity and $\mathbf{H}=<\beta >$ is the cyclic group of order $s(r+1).$ By the theorem of group homomorphism,

$$(\mathbf{H}\times \langle \alpha \rangle )/\langle \alpha \rangle \cong \mathbf{H}/(\mathbf{H}\cap \langle \alpha \rangle ),$$

Let $i_1,i_2,\dots ,i_t$ be t distinct elements, such that $0\le i_1<i_2<\cdots <i_t<2(r+1).$ Denote $I=\{i_1,i_2,\dots ,i_t\},A=i_1+i_2+\dots +i_t$ and $\mathbf{B}=\{{\beta }^{i_1},{\beta }^{i_2},\dots ,{\beta }^{i_t}\}$ be a set of coset representatives of $\mathbf{H}\times \langle \alpha \rangle$. Let

$$\overrightarrow{a}=(\alpha {\beta }^{i_1},\cdots ,{\alpha }^m{\beta }^{i_1},\alpha {\beta }^{i_2},\cdots ,{\alpha }^m{\beta }^{i_2},\cdots ,\alpha {\beta }^{i_t},\cdots ,{\alpha }^m{\beta }^{i_t}).$$

Similar with Theorem 1, we get

$$\begin{array}{ccc}\hfill L_{\overrightarrow{a}}\left({\beta }^z{\alpha }^k\right)& =& \prod _{1\le j\le m,j\ne k}({\beta }^z{\alpha }^k-{\beta }^z{\alpha }^j)\prod _{l\in I,l\ne z}\prod _{j=1}^m({\beta }^z{\alpha }^k-{\beta }^l{\alpha }^j)\hfill \\ & =& {\beta }^{z(m-1)}·m·{\alpha }^{-k}\prod _{l\in I,l\ne z}({\beta }^{zm}-{\beta }^{lm}.)\hfill \end{array}$$

Since ${\beta }^{s(r+1)}=1,$ then ${\beta }^{r+1}={\xi }_s,$ where ${\xi }_s$ is s-th primitive root of unity. So ${\beta }^r={\xi }_s{\beta }^{-1}.$ Let $v={\displaystyle \prod _{l\in I,l\ne z}}({\beta }^{zm}-{\beta }^{lm}).$ Since $s\mid m,$ then

$$\begin{array}{ccc}\hfill v^r& =& \prod _{l\in I,l\ne z}({\left({\beta }^{-1}\right)}^{zm}-{\left({\beta }^{-1}\right)}^{lm})\hfill \\ & =& \prod _{l\in I,l\ne z}{\beta }^{-(l+z)m}({\beta }^{lm}-{\beta }^{zm})\hfill \\ & =& {(-1)}^{t-1}{\beta }^{-(A+(t-2\left)z\right)m}v.\hfill \end{array}$$

So $v^{r-1}={(-1)}^{t-1}{\beta }^{-(A+(t-2\left)z\right)m}.$

Let g be a generator of ${\mathbb{F}}_q^{*}.$ It follows that $\beta =g^{\frac{r-1}{s}}$ and $-1=g^{\frac{r^2-1}{2}}.$ So

$$v=g^{\frac{(r+1)}{2}(t-1)-[A+(t-2)z]\frac{m}{s}}.$$

Case 1: If m odd and t even, we can take $\lambda =g^{\frac{(r+1)}{2}(t-1)-A·\frac{m}{s}}.$ Hence, we have $\lambda L_{\overrightarrow{a}}\left({\beta }^z{\alpha }^k\right)$ is square element of ${\mathbb{F}}_q^{*}.$

Case 2: If m even and $2\mid \frac{m}{s}$, we can take $\lambda =g^{\frac{(r+1)}{2}(t-1)}.$ Hence, we have $\lambda L_{\overrightarrow{a}}\left({\beta }^z{\alpha }^k\right)$ is square element of ${\mathbb{F}}_q^{*}.$

So there exists a q-ary MDS self-dual code with length $n$.  □

4. Conclusions

In this paper, based on the method from [9], we construct several classes of MDS self-dual code over finite fields with odd characteristics via the generalized Reed-Solomon code and extend the generalized Reed-Solomon code.