Explicit Constructions of MDS Self-Dual Codes

Sok, Lin

doi:10.1109/tit.2019.2954877

Cited by 27 publications

(14 citation statements)

References 21 publications

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“…For 1)-3), it can be easily checked that h ′ (α) is a square for any α ∈ U. See also [31,Theorem 2 and Theorem 4]. For 4)-6), it was already checked, in [32, Theorem 3 and Theorem 4], that h ′ (α) is a square for any α ∈ U .…”

Section: Construction Of One Dimensional Hullmentioning

confidence: 98%

MDS linear codes with one dimensional hull

Sok¹

2020

Preprint

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We define the Euclidean hull of a linear code C as the intersection of C and its Euclidean dual C ⊥ . The hull with low dimensions gets much interest due to its crucial role in determining the complexity of algorithms for computing the automorphism group of a linear code and checking permutation equivalence of two linear codes. It has been recently proved that any q-ary [n, k] linear code with q > 3 gives rise to a linear code with the same parameters and having zero dimensional Euclidean hull, which is known as a linear complementary dual code. This paper aims to explore explicit constructions of families of MDS linear codes with one dimensional Euclidean hull. We obtain several classes of such codes.

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Section: Construction Of One Dimensional Hullmentioning

confidence: 98%

MDS linear codes with one dimensional hull

Sok¹

2020

Preprint

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“…In [26], Zhang and Feng presented some new constructions of MDS Euclidean self-dual codes via cyclotomy. In [27], Sok gave some explicit constructions of MDS Euclidean self-dual codes via rational function fields. In the following Table I, we summary some known results about the construction of MDS Euclidean self-dual codes.…”

Section: Related Workmentioning

confidence: 99%

Construction of MDS Euclidean Self-Dual Codes via Two Subsets

Fang

Xia

2020

Preprint

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The parameters of a q-ary MDS Euclidean self-dual codes are completely determined by its length and the construction of MDS Euclidean self-dual codes with new length has been widely investigated in recent years. In this paper, we give a further study on the construction of MDS Euclidean self-dual codes via generalized Reed-Solomon (GRS) codes and their extended codes. The main idea of our construction is to choose suitable evaluation points such that the corresponding (extended) GRS codes are Euclidean self-dual.Firstly, we consider the evaluation set consists of two disjoint subsets, one of which is based on the trace function, the other one is a union of a subspace and its cosets. Then four new families of MDS Euclidean self-dual codes are constructed. Secondly, we give a simple but useful lemma to ensure that the symmetric difference of two intersecting subsets of finite fields can be taken as the desired evaluation set. Based on this lemma, we generalize our first construction and provide two new families of MDS Euclidean self-dual codes. Finally, by using two multiplicative subgroups and their cosets which have nonempty intersection, we present three generic constructions of MDS Euclidean self-dual codes with flexible parameters. Several new families of MDS Euclidean self-dual codes are explicitly constructed.

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“…The effort to find optimal codes has lasted for decades, and is still ongoing. To see the whole history, we refer to [3], [11]- [14], [37], [39].…”

Section: Introductionmentioning

confidence: 99%

Self-Dual Codes, Symmetric Matrices, and Eigenvectors

Kim

Choi

2021

IEEE Access

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We introduce a consistent and efficient method to construct self-dual codes over GF (q) using symmetric matrices and eigenvectors from a self-dual code over GF (q) of smaller length where q ≡ 1 (mod 4). Using this method, which is called a 'symmetric building-up' construction, we improve the bounds of the best-known minimum weights of self-dual codes with lengths up to 40, which have not significantly improved for almost two decades. We focus on a class of self-dual codes, which includes double circulant codes. We obtain 2967 new self-dual codes over GF (13) and GF (17) up to equivalence. Also, we compute the minimum weights of quadratic residue(QR) codes that were previously unknown. These are: [20,10,10] QR self-dual code over GF (23), [24,12,12] QR self-dual codes over GF (29) and GF (41), and [32,16,14] QR self-dual codes over GF (19). They have the highest minimum weights so far.INDEX TERMS eigenvectors, optimal codes, quadratic residue codes, self-dual codes, symmetric matrix, symmetric self-dual code

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Explicit Constructions of MDS Self-Dual Codes

Cited by 27 publications

References 21 publications

MDS linear codes with one dimensional hull

MDS linear codes with one dimensional hull

Construction of MDS Euclidean Self-Dual Codes via Two Subsets

Self-Dual Codes, Symmetric Matrices, and Eigenvectors

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