# Construction of MDS Self-dual Codes over Finite Fields

In this paper, we obtain some new results on the existence of MDS self-dual codes utilizing (extended) generalized Reed-Solomon codes over finite fields of odd characteristic. For some fixed q, our results can produce more classes of MDS self-dual codes than previous works.

## Authors

• 4 publications
• 1 publication
• 20 publications
• ### New Parameters on MDS Self-dual Codes over Finite Fields

In this paper, we produce new classes of MDS self-dual codes via (extend...
11/07/2018 ∙ by Xiaolei Fang, et al. ∙ 0

• ### New MDS Self-dual Codes over Finite Fields of Odd Characteristic

In this paper, we produce new classes of MDS self-dual codes via (extend...
11/07/2018 ∙ by Xiaolei Fang, et al. ∙ 0

• ### New MDS Self-dual Codes over Finite Fields

In this paper, we produce new classes of MDS self-dual codes via (extend...
11/07/2018 ∙ by Xiaolei Fang, et al. ∙ 0

• ### The Subfield Codes of [q+1, 2, q] MDS Codes

Recently, subfield codes of geometric codes over large finite fields (q)...
08/03/2020 ∙ by Ziling Heng, et al. ∙ 0

• ### Adelic Extension Classes, Atiyah Bundles and Non-Commutative Codes

This paper consists of three components. In the first, we give an adelic...
09/04/2018 ∙ by Lin Weng, et al. ∙ 0

• ### Some Generalizations of Good Integers and Their Applications in the Study of Self-Dual Negacyclic Codes

Good integers introduced in 1997 form an interesting family of integers ...
01/14/2018 ∙ by Supawadee Prugsapitak, et al. ∙ 0

• ### New MDS Symbol-Pair Codes from Repeated-Root Cyclic Codes over Finite Fields

Symbol-pair codes are introduced to guard against pair-errors in symbol-...
10/09/2020 ∙ by Junru Ma, et al. ∙ 0

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## 1 Introduction

A linear code over a finite field of length , dimension and minimum distance is called MDS (maximum distance separable) if it attains the Singleton bound: . MDS codes have been of much interest from many researchers due to their theoretical significant and practical implications, see [References], [References], [References].

For a linear code , we denote the dual of under Euclidean inner product by . The linear code is called self-dual if . Self-dual codes have attracted attention from coding theory, cryptograph and other fields. It has been found various applications in cryptography, in particular secret sharing scheme [References], [References], [References] and combinatorics [References].

More recently, the application of MDS codes renewed the interest in the construction of MDS self-dual codes, see [References], [References], [References], [References]. K. Guenda [References] constructed MDS Euclidean and Hermitian self-dual codes which are extended cyclic duadic codes or negacyclic codes. She also constructed Euclidean self-dual codes which are extended negacyclic codes.

Generalized Reed-Solomon () codes is a class of MDS code which has nice algebraic structure. It can be systematically constructed and has been found wide applications in practice. MDS self-dual codes through codes have been studied by L. Jin and C. Xing [References], where they constructed several classes of MDS self-dual codes through codes by choosing suitable parameters. In [References], H. Yan generalizes the technique in [References] and construct several classes of MDS self-dual codes via codes and extended codes.

Since MDS self-dual codes over finite field of even characteristic with any possible parameter have been found in [References]. In this paper, we obtain some new results on the existence of MDS self-dual codes through (extended) codes over finite fields of odd characteristic. Some results in this paper extend those of [References] and [References]. Comparing to previous works, for some fixed square prime power , our construction will produce more -ary MDS self-dual codes.

This paper is organized as follows. In Section 2 we will introduce some basis knowledge and auxiliary results on codes and extended codes. In particular, Corollary 2.1 and Corollary 2.2 give a criterion for an (extended) code to be self-dual. In Section 3 we will present our main results on the construction of MDS self-dual codes. Our main tools are Corollary 2.1 and Corollary 2.2. We choose suitable parameters to make the conditions in Corollary 2.1 and Corollary 2.2 hold.

## 2 Generalized Reed-Solomon codes

In this section, we introduce some basic notations and results on generalized Reed-Solomon code. Throughout this paper, let be a finite field with elements, and let be a positive integer with . Choose to be an -tuple of distinct elements of . Put with . For an integer with , then linear code

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

is the generalized Reed-Solomon or code.
The code has a generator matrix

It is well-known that the code is a -ary -MDS code and its dual is also MDS.

We define

 La(αi)=∏1≤j≤n,j≠i(αi−αj). (2)

The dual of code is explicitly determined. Precisely, let

be the all-one vector with appropriate length. The dual of

is , where with for .

###### Remark 2.1.

The matrix is a generator matrix of , and is a generator matrix of . It is clear that

 Gk(a,v)=Gk(a,1)diag(v1,v2,…,vn),

where is the diagonal matrix with diagonal entries .

We now introduce some basic notations and results on extended generalized Reed-Solomon code. The -dimensional extended code of length given by

 GRSk(a,v,∞)={(v1f(α1),…,vn−1f(αn−1),fk−1):f(x)∈Fq[x],deg(f(x))≤k−1}, (3)

where is the coefficient of in .
Clearly, the code has a generator matrix

 Gk(a,v,∞)=⎛⎜ ⎜ ⎜ ⎜ ⎜⎝v1v2…vn−10v1α1v2α2…vn−1αn−10⋮⋮⋱⋮⋮v1αk−11v2αk−12…vn−1αk−1n−11⎞⎟ ⎟ ⎟ ⎟ ⎟⎠.

It is known that is -ary -MDS code and its dual is also MDS. Precisely, the dual code of is , where with for .

###### Remark 2.2.

Let be a generator matrix for , and let be a generator matrix for . It is easy to see that the relationship between , and is follows

where is the diagonal matrix with diagonal entries .

The following lemma gives a criterion for a code to be self-dual.

###### Lemma 2.1.

Let be an even integer, and . The code is a -ary MDS self-dual code over if and only if for any codeword , where , we have .

###### Proof.

Since , for any , we have , where is a generator matrix for . By remark(2.1), we can obtain that

 Gk(a,v)cT=(Gk(a,1)⋅diag(v1,v2,…,vn))⋅cT=Gk(a,1)⋅(v21f(α1),…,v2nf(αn))T=0.

This implies that is MDS self-dual if and only if

 Gk(a,1)⋅(v21f(α1),…,v2nf(αn))T=0.

In other words, . Therefore, the lemma is proved. ∎

###### Corollary 2.1.

([References], Corollary 2.4) For an even integer , and . If there exist such that for some for all , then the code defined in (1) is MDS self-dual, where for all .

The following lemma is important and it gives the necessary and sufficient condition to construct self-dual codes via the extended codes.

###### Lemma 2.2.

Let be even and . The code is a -ary MDS self-dual code over if and only if for any codeword , where , we have .

###### Proof.

Since is a -ary MDS self-dual code , we have , where is a generator matrix for , with . By remark(2.2), we get

 Gk(a,v,∞)cT=(Gk(a,1,∞)diag(v1,v2,…,vn−1,1))⋅cT=Gk(a,v,∞)⋅(v21f(α1),…,v2n−1f(αn−1),fk−1)T=0.

It follows that is a -ary MDS self-dual code over if and only if

 Gk(a,1,∞)(v21f(α1),…,v2n−1f(αn−1),fk−1)T=0.

That means . This completes the proof.

###### Corollary 2.2.

([References], Lemma 2) Let be even and . If for some for all , then the code defined in (3) is MDS self-dual, where for all .

## 3 Main Results

The existence of MDS self-dual codes over finite fields of even characteristic has been completely addressed in [References]. In this section, we construct serval new classes of MDS self-dual codes, by using codes and extended codes over finite fields of odd characteristic.
The following lemma can be found in [References].

###### Lemma 3.1.

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.□

Let denote the set of nonzero squares of .

###### Theorem 1.

Let where is odd prime power, and let be a positive integer with and . Assume both and are even. Then there exists a -ary MDS self-dual code over .

###### Proof.

Let be a primitive -th root of unity. Since one has group isomorphism and embedding

 F∗r/(F∗r∩⟨α⟩)≃(F∗r×⟨α⟩)/⟨α⟩≤F∗q/⟨α⟩,

and , we can choose () to be cost representatives of and

 a=(αβ0,…,αmβ0,αβ1,…,αmβ1,…,αβt−1,…,αmβt−1).

Then the entries of are distinct elements of . Note that

 xm−γm=∏1≤j≤m(x−γαj)

for any . By Lemma 3.1, for any and for any , we have

 La(βzαi)=∏1≤j≤m,j≠i(βzαi−βzαj)×∏t−1l=0,l≠z∏mj=1(βzαi−βlαj)=βzm−1⋅m⋅α−i⋅∏t−1l=0,l≠\parz(βzm−βlm).

Note that since .

Since is even and is primitive -th root of unity, then . As a consequence, . By Corollary 2.1, there exists a -ary MDS self-dual code over .

###### Remark 3.1.

Theorem 3.4 (i) in [References] is a special case of the preceding result when .

###### Example 3.1.

When and , we can choose and . In this case and the preceding result shows that there exists -ary MDS self-dual code of length which, as our best knowledge, has not been found in previous references.

###### Theorem 2.

Let where is odd prime power, and let be a positive integer with and . Assume is odd. Then there exists a -ary MDS self-dual code over .

###### Proof.

The proof is similar as that of Theorem 1.

 La(βzαi)=βzm−1⋅m⋅α−i⋅∏t−1l=0,l≠z(βzm−βlm).

Note that since .

Since is odd, then . Therefore . By Corollary 2.2, there exists a -ary MDS self-dual code over .

###### Theorem 3.

Let where is odd prime power, and let be a positive integer with and . Assume is even. Then there exists a -ary MDS self-dual code over .

###### Proof.

The elements and are chosen in the same way as in Theorem 1. We define the generalized Reed-Solomn code with

 a=(0,αβ0,…,αmβ0,αβ1,…,αmβ1,…,αβt−1,…,αmβt−1).

Similar as the above proofs, for any and for any ,

 La(βzαi)=βzαi×∏1≤j≤m,j≠i(βzαi−βzαj)×∏t−1l=0,l≠z∏1≤j≤m(βzαi−βlαj)=βzm⋅m⋅∏t−1l=0,l≠z(βzm−βlm)

and

 La(0)=t−1∏l=0m∏j=1(0−βlαj)=(t−1∏l=0βl)m⋅αm(m+1)2=±(t−1∏l=0βl)m.

Note that , and . Hence both and are in . By Corollary 2.2, there exists a -ary MDS self-dual code of length . ∎

###### Theorem 4.

Let with odd prime. For any with and , if is even, then there exists self-dual MDS code with length .

###### Proof.

Let be an -dimensional -vector subspace in satisfying . Denote by a primitive element of order . Choose

 a=2t−1⋃j=0(ωj+V).

For any ,

 (4)

where the last equality follows from that and runs through when runs through .

Denote by . Then . Since is even, we can deduce that . As a consequence, which is independent of the choice of . In this case we choose . Then . By Corollary 2.1, there exists self-dual MDS code with length . ∎

###### Remark 3.2.

For , the preceding result is exactly Theorem 4 (ii) in [References].

In the case is a square, usually Theorems 1-4 will present more classes of MDS self-dual codes than the previous results.

###### Example 3.2.

For , there are 243 different for which MDS self-dual code of lengths are constructed in all the previous works(in Table 1). In our constructions (Theorems 1-4), there are 713 different classes of MDS self-dual codes of different lengths.

## Acknowledgements

This work is supported by the self-determined research funds of CCNU from the colleges’ basic research and operation of MOE (Grant No. CCNU18TS028). The work of J.Luo is also supported by NSFC under Grant 11471008.

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