 # New Parameters on MDS Self-dual Codes over Finite Fields

In this paper, we produce new classes of MDS self-dual codes via (extended) generalized Reed-Solomon codes over finite fields of odd characteristic. Among our constructions, there are some new parameters on MDS self-dual code which have never been reported.

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## 1 Introduction

Let be the finite field with elements, where is a prime power. A linear code of length , dimension and minimum distance over is usually called a -ary code. If the parameters of the code attach the Singleton bound: , the code is called a maximum distance separable (MDS) code. MDS codes are widely applied in various occasions due to their nice properties, see [References, References, References].

The dual code of a linear code in , denoted by , is the linear subspace of , which is orthogonal to . If , is called a self-dual code. Self-dual codes have important applications in coding theory [References], cryptograph [References, References], combinatorics [References, References] and other related areas.

MDS self-dual codes have good properties due to its optimality with respect to the Singleton bound and its self-duality, which have attracted a lot of attentions in recent years. Some people constructed MDS self-dual codes through the way of orthogonal designs, see [References, References, References, References]. Some people utilized constacyclic codes to construct MDS self-dual codes, see [References, References, References, References]. Some people make the construction by Reed-Solomon codes, generalized Reed-Solomon codes or extended generalized Reed-Solomon codes, see [References, References, References, References, References].

Parameters of MDS self-dual codes are completely determined by their lengths , that is, . Therefore, the problem for constructing different MDS self-dual codes can be transformed to find MDS self-dual codes with different lengths. In [References] Grassl and Gulliver showed that the problem has been completely solved over the finite fields of characteristic . But the constructions on the finite fields of odd characteristic is still far from complete. For example, if , more than 3000 MDS self-dual codes with different even lengths possibly exist assuming MDS conjecture is valid. But up to now, only 702 MDS self-dual codes of different even lengths are able to be constructed. In [References], Jin and Xing constructed some classes of new MDS self-dual codes through generalized Reed-Solomon codes. In [References], Yan generalized the technique in [References] and constructed several classes of MDS self-dual codes via generalized Reed-Solomon codes and extended generalized Reed-Solomon codes. In [References], Labad, Liu and Luo produced more classes of MDS self-dual codes based on [References] and [References]. All the known results on the construction of MDS self-dual codes are depicted in Table 1. Some other constructions, using building up technique can be found in [References]. Based on [References], [References] and [References], in this paper, we give more constructions on MDS self-dual code. Among our constructions, there are several MDS self-dual codes with new parameters (see Table 2). In particular, for square , we can produce much more MDS self-dual codes than previous work.

This paper is organized as follows. In Section 2, we will introduce some basic knowledge and useful results on (extended) generalized Reed-Solomon codes. In Section 3, we will present our main results on the constructions of MDS self-dual codes. In Section 4, we will make a conclusion.

## 2 Preliminaries

In this section, we introduce some basic notations and useful results on (extended) generalized Reed-Solomon codes (or (extended) codes for short). Readers are referred to [References, Chapter 10] for more details.

Let be the finite field with elements and be any integer with . Choose two -tuples and , where , ( may not be distinct) and , are distinct elements in . For an integer with , the code of length associated with and is defined below:

 GRSk(→a,→v)={(v1f(α1),…,vnf(αn)):f(x)∈Fq[x],deg(f(x))≤k−1}. (1)

It is well-known that the code is a -ary -MDS code and its dual code is also an MDS code [References, Chapter 11].

We define

 L→a(αi)=∏1≤j≤n,j≠i(αi−αj).

Let denote the set of nonzero squares of . The following result is useful in our constructions and it has been shown in [References].

###### Lemma 2.1.

([References], Corollary 2.4) Let be an even integer and . If there exists such that for all , then there exists with such that the code defined in (1) is an MDS self-dual code of length .

Moreover, extended code can also be applied into the construction of MDS self-dual codes. We can choose the two -tuples and in the same way as code. Then the extended code of length associated with and is defined as follows:

 GRSk(→a,→v,∞)={(v1f(α1),⋯,vn−1f(αn−1),fk−1):f(x)∈Fq[x],deg(f(x))≤k−1}, (2)

where is the coefficient of in .

It is also well-known that is a -ary -MDS code and its dual is also an MDS code [References, Chapter 11].

###### Lemma 2.2.

([References], Lemma 2) Let be an even integer and . If for all , then there exists with such that the code defined in (2) is an MDS self-dual code of length .

###### Lemma 2.3.

([References], Lemma 3) Let be a positive integer and let be a primitive -th root of unity. Then for any , we have

 ∏1≤j≤m,j≠i(αi−αj)=mα−i.

## 3 Main Results

In this section, we will give several new constructions of MDS self-dual codes utilizing the additive group structure on and the multiplicative group structure of .

###### Theorem 1.

Let , where is an odd prime power. Suppose . For , assume is even.

(i). If is even and , there exists a -ary MDS self-dual code.

(ii). Assume . There exists a -ary MDS self-dual code except that is even, is even and .

###### Proof.

Let be a primitive -th root of unity in and be the cyclic group of order . By the second fundamental theorem of group homomorphism, we have

 S/(S∩⟨α⟩)≃(S×⟨α⟩)/⟨α⟩≤F∗q/⟨α⟩.

(i). We choose distinct elements such that . Denote and . Let be a set of coset representatives of and

 →a=(αβi1,…,αmβi1,αβi2,…,αmβi2,⋯αβit,…,αmβit).

Then the entries of are distinct in . We will show that there exists such that is an MDS self-dual code of length .

Note that . By Lemma 2.3, for any and , we deduce

 \par\parL→a(βzαk)=∏1≤j≤m,j≠k(βzαk−βzαj)⋅∏l∈I,l≠zm∏j=1(βzαk−βlαj)=βz(m−1)⋅m⋅α−k⋅∏l∈I,l≠z(βzm−βlm).

Let . Then

 \par\parur=∏l∈I,l≠z(β−zm−β−lm)=∏l∈I,l≠zβ−(l+z)m(βlm−βzm)=(−1)t−1⋅β−(∑l∈I,l≠zl+(t−1)z)m⋅u=(−1)t−1⋅β−(A+(t−2)z)m⋅u.

So .

Let be a generator of such that and . So

 ur−1=gr2−12⋅(t−1)⋅g−(r−1)⋅(A+(t−2)z)m.

It follows that

 u=gr+12⋅(t−1)−(A+(t−2)z)m+i(r+1)forsomei.

Note that . We take . Since is even, we obtain that . Choose with . Define

 →v=(vi1,1,⋯,vi1,m,⋯,vit,1,⋯,vit,m).

Then by Lemma 2.1, is an MDS self-dual code.

Therefore, we know that there exists a -ary MDS self-dual code with the length of .

(ii). Similarly as (i), we let

 →a=(0,αβi1,⋯,αmβi1,αβi2,⋯,αmβi2,⋯αβit,⋯,αmβit).

We will find such that is an MDS self-dual code of length .

For any and for any , , we have

 L→a(βzαk)=βzαk⋅∏1≤j≤m,j≠k(βzαk−βzαj)⋅∏l∈I,l≠zm∏j=1(βzαk−βlαj)=βzm⋅m⋅∏l∈I,l≠z(βzm−βlm)

and

 L→a(0)==∏l∈Im∏j=1(0−βlαj)=(−1)m⋅αmt2⋅(∏l∈Iβl)m=±(∏l∈Iβl)m.

Denote . We can obtain that for some , in the same way as (i). We discuss in the following.

: If is odd and is even, we have is even. It follows that .

: If is even and , we can choose such that is an even integer. It follows that is even. Hence .

: If is even, is odd and , we can choose such that is an odd integer. It follows that is even. Hence .

Note that . So in all three cases, one has .

It is easy to verify that . We choose and , with . Define

 →v=(v0,vi1,1,⋯,vi1,m,⋯,vit,1,⋯,vit,m).

Then by Lemma 2.2, is an MDS self-dual code with the length of , except that is even, is even and .

###### Example 3.1.

Let , , and . Then . It is easy to verify that is even. By Theorem 1 (i), we know there exists the MDS self-dual code of length . This is a new parameter of MDS self-dual code which was not reported in any previous work.

###### Theorem 2.

Let , where is an odd prime power. Suppose . For , we assume is odd and . Then there exists a -ary MDS self-dual code over .

###### Proof.

The elements and are the same as in Theorem 1 (i). We choose distinct even elements , where . We take and . It is easy to see that is even. Let

 →a=(αβi1,…,αmβi1,αβi2,…,αmβi2,⋯αβit,…,αmβit).

The proof is similar as in Theorem 1 (i). We deduce that

 L→a(βzαk)=βz(m−1)⋅m⋅α−k⋅∏l∈I,l≠z(βzm−βlm).

Let . We can obtain in the same way as Theorem 1 (i). Since is odd, and are even, then is even. It follows that .

Since is odd, then is even, that is . Note that . Therefore, we have . Choose , with . Define

 →v=(vi1,1,⋯,vi1,m,⋯,vit,1,⋯,vit,m).

By Lemma 2.2, is an MDS self-dual code with the length of . ∎

###### Example 3.2.

Let , , and . Then . It is easy to verify that . By Theorem 2, we know there exists MDS self-dual code of length . This is a new parameter of MDS self-dual code which has not been covered by previous work.

###### Theorem 3.

Let , where is an odd prime power. Let , be even, and . Assume that .

(i). Assume . If and are even, there exists a -ary MDS self-dual code.

(ii). Assume . There exists a -ary MDS self-dual code.

###### Proof.

We choose the elements and in the same way in Theorem 1 (i). Here is a primitive -th root of unity and is a primitive -th root of unity in . Let . From the second fundamental theorem of group homomorphism, we get

 S/(S∩⟨α⟩)≃(S×⟨α⟩)/⟨α⟩≤F∗q/⟨α⟩.

(i). We choose distinct elements such that and denote .

Let be a set of coset representatives of and

 →a=(αβi1,⋯,αmβi1,αβi2,⋯,αmβi2,⋯αβit,⋯,αmβit).

Then the entries of are distinct elements of . We will show that there exists such that is an MDS self-dual code of length .

Similarly as Theorem 1 (i), we get

 L→a(βzαk)=∏1≤j≤m,j≠k(βzαk−βzαj)⋅∏l∈I,l≠zm∏j=1(βzαk−βlαj)=βz(m−1)⋅m⋅α−k⋅∏l∈I,l≠z(βzm−βlm).

Since the order of is , then is a primitive -th root of unity. So .

Let . Since , then

 ur=∏l∈I,l≠z(βzm−βlm)=u,

which implies .

If both and are even, one has .

Now we obtain . Hence .

Choose with . Define

 →v=(vi1,1,⋯,vi1,m,⋯,vit,1,⋯,vit,m).

Then by Lemma 2.1, is an MDS self-dual code.

Therefore, we know that there exists a -ary MDS self-dual code with the length of .

(ii). Similarly as (i), we let

 →a=(0,αβi1,…,αmβi1,αβi2,…,αmβi2,⋯αβit,…,αmβit).

We will find such that is an MDS self-dual code of length .

For any and for any , one has

 L→a(βzαk)=βzαk⋅∏1≤j≤m,j≠k(βzαk−βzαj)⋅∏l∈I,l≠zm∏j=1(βzαk−βlαj)=βzm⋅m⋅∏l∈I,l≠z(βzm−βlm)

and

 L→a(0)=∏l∈Im∏j=1(0−βlαj)=αm(m+1)2⋅(∏l∈Iβl)m=±(∏l∈Iβlm).

Since the order of is , then , where is a primitive -th root of unity. So with . It implies that . Therefore, we know that . Since , then . We choose and , with . Define

 →v=(v0,vi1,1,⋯,vi1,m,⋯,vit,1,⋯,vit,m).

Then by Lemma 2.2, is an MDS self-dual code with the length of .

###### Example 3.3.

Let , , and . Then both and are even. Note that , by Theorem 3 (i), we know there exists MDS self-dual code of length . This MDS self-dual code has not been reported in any previous references.

###### Theorem 4.

Let , where is an odd prime and is a positive integer. Then there exists a -ary MDS self-dual code of length of , where .

###### Proof.

Denote by . Let be an -dimensional

-vector subspace of

, with . Choose , such that . Let , and .