 # New 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 many new parameters of MDS self-dual code which have never been reported. For large square q, we can produce more than approximately 50% times of new MDS self-dual codes with different lengths than the previous results.

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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: , then 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 a 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, 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 their self-duality, which have attracted a lot of attention in recent years. There are various ways to construct MDS self-dual codes. They mainly are: (1). orthogonal designs, see [References, References, References]; (2). building up technique, see [References, References]; (3). constacyclic codes, see [References, References, References]; (4). (generalized and/or extended) Reed-Solomon codes, see [References. References, References, References, References, References, References].

Parameters of MDS self-dual codes are completely characterized 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 of MDS self-dual codes on finite fields of odd characteristic are 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 (MDS conjecture says that the length of nontrivial -ary MDS code with odd prime power, is bounded by ). But up to now, only 702 -ary MDS self-dual codes of different even lengths have been 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 systematic constructions of MDS self-dual codes are depicted in Table 1.

Based on [References], [References] and [References], we give more constructions of MDS self-dual codes in this paper. 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 works.

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 notation 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 an 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 as follows:

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

The code is a -ary MDS code and its dual is also MDS [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 to the construction of MDS self-dual codes. For and , 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 . The code is a -ary MDS code and its dual is also MDS [References, Chapter 11].

We present another two useful results, which have been shown in [References].

###### 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 ,

 ∏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 multiplicative group structure of and the additive group structure on .

###### Theorem 1.

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

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

(ii). if , then there exists a -ary MDS self-dual code except the case 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). Let be a set of coset representatives of with . Denote by , and

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

Obviously, 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 . We calculate

 \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 . Then

 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).

By Lemma 2.1, is an MDS self-dual code. Therefore, there exists a -ary MDS self-dual code with length .

(ii). As in (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 , ,

 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

Denote . We obtain for some , in the same way as (i). The following cases are considered.

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

: If is even and , we can choose such that is even. 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 . As a result, one always has .

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

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

By Lemma 2.2, is an MDS self-dual code with length , except the case that is even, is even and .

###### Example 3.1.

Let , , and . Then . By Theorem 1, there exists MDS self-dual code of length . This is a new parameter of MDS self-dual code.

###### Theorem 2.

Let , where is an odd prime power. Suppose . If , is odd and , then there exists a -ary MDS self-dual code over .

###### Proof.

Recall and in the proof of Theorem 1 (i). Choose with and () even. Denote by distinct and

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

The main goal is to find such that is an MDS self-dual code. Similarly as in Theorem 1 (i), for , and , 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, it follows that is even which implies .

Since is odd, it implies that . Note that . Therefore, . Choose , with . Define

 →v=(vi1,1,…,vi1,m,…,vit,1,…,vit,m).

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

###### Example 3.2.

If , , and , then . By Theorem 2, there exists an 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 , even, and . For ,

(i). if , both and are even, then there exists a -ary MDS self-dual code.

(ii). if , then there exists a -ary MDS self-dual code.

###### Proof.

Let be a primitive -th root of unity and be a primitive -th root of unity in . Let . From the second fundamental theorem of group homomorphism,

 S/(S∩⟨α⟩)≃(S×⟨α⟩)/⟨α⟩≤F∗q/⟨α⟩.

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

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

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

Similarly as Theorem 1 (i),

 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).

Note that the order of is . Then is a primitive -th root of unity and . Let . Since , it follows that

 ur=∏l∈I,l≠z(βzm−βlm)=u,

which implies . If both and are even, then . Now we obtain . Hence . Choose with . Define

 →v=(vi1,1,…,vi1,m,…,vit,1,…,vit,m).

According to Lemma 2.1, is an MDS self-dual code with length .

(ii). As in (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).

The order of is , which implies that since . Therefore, . Since , . We choose and , with . Define

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

According to Lemma 2.2, is an MDS self-dual code with length .

###### Example 3.3.

If , , , and , then both and are even. Note that . By Theorem 3, there exists a -ary 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. There exists a -ary MDS self-dual code of length , where .

###### Proof.

Denote by . Let be an -dimensional

-vector subspace of

, with . Choose , such that . Let , and . A routine calculation shows that

 L→a(ak0,j0)= ∏1≤k,j≤pe(k,j)≠(k0,j0)(ak0,j0−ak,j) =