 # Two Classes of New MDS Self-dual Codes over Finite Fields

In this paper, we produce two new classes of MDS self-dual codes via generalized Reed-Solomon (GRS) codes over finite fields. Among our constructions, for large square q, we can produce about 1/8· q new MDS self-dual codes with different lengths

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## 1 Introduction

Let be a prime power and be a finite field of cardinality . A linear code over , denote by is a linear -subspace of with dimension and minimal (Hamming) distance . If the parameters can reach the Singleton bound, that is , then we call be a maximum distance separable (MDS) code. Denote the (Euclidean) dual code of by . If , then is called a self-dual code.

Both MDS codes and self-dual codes have theoretical interest and practical importance. Thus the study of MDS self-dual codes have attracted a lot of interest, especially for the construction of MDS self-dual codes. MDS self-dual codes can be constructed in various ways, which 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].

It is noted that the parameters of MDS self-dual codes can be completely determined by the code length . For the finite field of even characteristic, Grassl and Gulliver completely determined the MDS self-dual codes of length for all in [References]. In [References, References, References], the authors obtained some new MDS self-dual codes through cyclic, constantcyclic and negacyclic codes. In [References], Jin and Xing firstly presented a general and efficient method to construct MDS self-dual codes by utilizing generalized Reed-Solomn (GRS) codes. Afterwards, many researchers start to construct MDS self-dual codes via GRS codes. In [References], Zhang and Feng gave a non-existence result, that is, when and , we can’t construct MDS self-dual code of length . We list all the known results on the systematic constructions of MDS self-dual codes, which are depicted in Table 1.

In odd characteristic case, the length of MDS self-dual codes is restricted among the above known results. For instance, in Row 7 of Table 1 (see [References]), the length must satisfy the restrictive conditions “ is even and ”. So it can produce a few MDS self-dual codes with different lengths. In this paper, we produce approximately many -ary MDS self-dual codes with different lengths.

## 2 Preliminaries

In this section, we introduce some basic notations and useful results on self-dual codes and GRS codes.

For two vectors

and of , we can define their (Euclidean) inner product:

 ⟨→b,→c⟩=b1c1+b2c2+⋯+bncn.

Let be a linear code over . The (Euclidean) dual of is defined as follows:

 C⊥={→u∈Fnq|for\, all→c∈C,⟨→u,→c⟩=0}.

If , then is called a self-dual code.

For , we choose two -tuples and , where , ( may not be distinct) and , are distinct elements in . Then the GRS code of length associated with and is defined below:

 GRSk(→a,→v)={(v1f(a1),…,vnf(an)):f(x)∈Fq[x],deg(f(x))≤k−1}, (1)

where .

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(ai)=∏1≤j≤n,j≠i(ai−aj).

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 a -ary MDS self-dual code of length .

The next lemma has been shown in [References] and is very useful for the proof of our main results.

###### Lemma 2.2.

([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 give our main results.

###### Theorem 1.

Let , where is an odd prime power and . For any , is even and , assume . There exists a -ary MDS self-dual code.

###### Proof.

Let be a primitive element of , , and . Denote by and and choose

 →a=(⟨α⟩,β⟨α⟩,…,βs−1⟨α⟩,γ⟨β⟩,γ3⟨β⟩,…,γ2(t−1)+1⟨β⟩).

Since , it follows that and . Therefore, for all and . By Lemma 2.2, it is necessary to consider two cases.

• For and ,

 L(βiαj) =r−2∏k=0,k≠j(βiαj−βiαk)⋅s−1∏l=0,l≠ir−2∏k=0(βiαj−βlαk)⋅t−1∏l=0r∏k=0(βiαj−βkα2l+1) (2) =βi(r−2)⋅(r−1)⋅α−j⋅s−1∏l=0,l≠i(β−2i−β−2l)⋅t−1∏l=0(αj(r+1)−γ(2l+1)(r+1)).

Note that and . We only need to consider . Then

 ur =s−1∏l=0,l≠i(β2i−β2l)=s−1∏l=0,l≠i(β2(i+l)⋅(β−2l−β−2i)) =β2((s−1)i+s−1∑l=0,l≠il)⋅(−1)s−1⋅s−1∏l=0,l≠i(β−2i−β−2l) =(−1)s−1⋅β2((s−2)i+s(s−1)2)⋅u.

So

 ur−1=(−1)s−1⋅β2((s−2)i+s(s−1)2)=g(r+1)(s−1)2⋅(r−1)+2((s−2)i+s(s−1)2)⋅(r−1),

that is

 u=g(r+1)(s−1)2+2((s−2)i+s(s−1)2)+k⋅(r+1) (3)

with some integer .

• For and ,

 L(γ2i+1βj)= r∏k=0,k≠j(γ2i+1βj−γ2i+1βk)⋅t−1∏l=0,l≠ir∏k=0(γ2i+1βj−γ2l+1βk) (4) ⋅s−1∏k=0r−2∏l=0(γ2i+1βj−αlβk) = (γr(2i+1)⋅(r+1)⋅β−j)⋅t−1∏l=0,l≠i(γ(2i+1)(r+1)−γ(2l+1)(r+1)) ⋅((−1)s⋅s−1∏k=0(β−2j+β−2k)).

We consider . Then

 ur =s−1∏k=0(β2j+β2k)=s−1∏k=0β2(j+k)(β−2j+β−2k) =β2(sj+s−1∑k=0k)⋅s−1∏k=0(β−2j+β−2k) =β2(sj+s(s−1)2)⋅u.

So

 ur−1=g(r−1)⋅2(sj+s(s−1)2),

that is , for some integer .

We choose .

• In (2), we have known that and are all nonzero square elements. From (3) and is even, we have is a nonzero square element. So .

• In (4), from the above analysis, it is clear that all elements except are nonzero square elements. So .

In summary, by Lemma 2.1, there exists a -ary MDS self-dual code over . ∎

###### Remark 3.1.

In Theorem 1, the number of is and the number of is . Thus the number of MDS self-dual codes is .

###### Example 3.1.

For , we can construct different for which MDS self-dual code of length by utilizing all the previous results (in Table 1). But we can produce different classes of MDS self-dual codes of different lengths by Theorem 1 in this paper. So the results in this paper can construct much more MDS sel-dual codes than previous works for large square .

###### Theorem 2.

Let , where is an odd prime power and . For any , and is odd, assume . There exists a -ary MDS self-dual code.

###### Proof.

Let be a primitive element of , , and . Denote by and and choose

 →a=(⟨β⟩,α⟨β⟩,…,αs−1⟨β⟩,γ⟨α⟩,γ3⟨α⟩,…,γ2(t−1)+1⟨α⟩)

Since , it follows that and . Therefore, for all and . By Lemma 2.2, it is necessary to consider two cases.

• For and ,

 L(αiβj) =r∏k=0,k≠j(αiβj−αiβk)⋅s−1∏l=0,l≠ir∏k=0(αiβj−αlβk)⋅t−1∏k=0r−2∏l=0(αiβj−αlγ2k+1) (5) =αir⋅(r+1)⋅β−j⋅s−1∏l=0,l≠i(αi(r+1)−αl(r+1))⋅t−1∏k=0(βj(r−1)−γ(2k+1)(r−1)).

Note that and . We only need to consider . Let . Then . Therefore,

 ur =t−1∏k=0(ξ−2j−ξ−(2k+1))=t−1∏k=0(ξ−2j−(2k+l)⋅(ξ2k+1−ξ2j)) =ξ−2tj−t−1∑k=0(2k+1)⋅(−1)t⋅t−1∏k=0(ξ2j−ξ2k+1) =(−1)t⋅ξ−2tj−t2⋅u.

So

 ur−1=(−1)t⋅ξ−2tj−t2=g(r+1)t2⋅(r−1)+(−2tj−t2)⋅(r−1)22,

that is,

 u=g(r+1)t2−(t2+2tj)⋅r−12+k⋅(r+1), (6)

for some integer .

• For and ,

 L(γ2i+1αj)= r−2∏k=0,k≠j(γ2i+1αj−γ2i+1αk)⋅t−1∏l=0,l≠ir−2∏k=0(γ2i+1αj−γ2l+1αk) (7) ⋅s−1∏k=0r∏l=0(γ2i+1αj−βlαk) = (γ(r−2)(2i+1)⋅(r−1)⋅α−j)⋅t−1∏l=0,l≠i(γ(2i+1)(r−1)−γ(2l+1)(r−1)) ⋅s−1∏k=0(−αj(r+1)−αk(r+1)).

Let and . Then Therefore,

 ur =t−1∏l=0,l≠i(ξ−(2i+1)−ξ−(2l+1))=t−1∏l=0,l≠iξ−(2i+1)−(2l+1)⋅(ξ2l+1−ξ2i+1) =(−1)t−1⋅ξ−(t−1)(2i+1)−t−1∑l=0,l≠i(2l+1)⋅t−1∏l=0,l≠i(ξ2i+1−ξ2l+1) =(−1)t−1⋅ξ−(t−2)(2i+1)−t2⋅u.

It follows that

 ur−1=g(r−1)⋅((r+1)(t−1)2−r−12⋅((t−2)(2i+1)+t2)),

that is with some integer . Since and is odd, then it is easy to verify that is even, which yields .

We choose .

• In (5), we have known that and are all nonzero square elements. From (6), and is odd, we obtain is a nonzero square element. So .

• In (7), from the above analysis, it is clear that all elements except are nonzero square elements. So .

In summary, by Lemma 2.1, there exists a -ary MDS self-dual code over . ∎

###### Remark 3.2.

In Theorem 2, the number of is and the number of is . Thus the number of MDS self-dual codes is .

###### Example 3.2.

For , we can construct different for which MDS self-dual code of length by utilizing all the previous results (in Table 1). But we can produce about different classes of MDS self-dual codes of different lengths by Theorem 2 in this paper. So the results in this paper can construct much more MDS sel-dual codes than previous works for large square .

## 4 Conclusion

Utilizing GRS codes, we construct two new classes of MDS self-dual codes. For any large square , we can produce about new MDS self-dual codes with different lengths. The number is already very large, but there is still a long way to go to reach the level of . So there are still a lot of work waiting to do.

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