Hulls of cyclic and negacyclic codes over finite fields

Sangwisut, Ekkasit; Jitman, Somphong; Ling, San; Udomkavanich, Patanee

doi:10.1016/j.ffa.2014.12.008

Cited by 42 publications

(46 citation statements)

References 12 publications

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“…On the other hand, the first s coordinates of Eq. (8) imply that εβu i f q (a i ) = u i g(a i ) = u i εf q (a i ), for any 1 ≤ i ≤ s. It follows from β = 1 and εu i = 0 that f q (a i ) = 0, i.e., f (a i ) = 0. Thus…”

Section: By Lemma 3mentioning

confidence: 98%

Euclidean and Hermitian Hulls of MDS Codes and Their Applications to EAQECCs

Fang

et al. 2020

IEEE Trans. Inform. Theory

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In this paper, we construct several classes of maximum distance separable (MDS) codes via generalized Reed-Solomon (GRS) codes and extended GRS codes, which we can determine the dimensions of their Euclidean hulls or Hermitian hulls. It turns out that the dimensions of Euclidean hulls or Hermitian hulls of the codes in our constructions can take all or almost all possible values. As a consequence, we can apply our results to entanglement-assisted quantum error-correcting codes (EAQECCs) and obtain several families of MDS EAQECCs with flexible parameters. The required number of maximally entangled states of these MDS EAQECCs can take all or almost all possible values. In particular, all possible parameters for the q-ary MDS EAQECCs of length n ≤ q are completely determined and several new classes of q-ary MDS EAQECCs of length n > q are also obtained.

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Section: By Lemma 3mentioning

confidence: 98%

Euclidean and Hermitian Hulls of MDS Codes and Their Applications to EAQECCs

Fang

et al. 2020

IEEE Trans. Inform. Theory

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“…A linear code C is said to be Hermitian self-dual (resp., Hermitian complementary dual) if C = Hull H (C) = C ⊥ H (resp., Hull H (C) = {0}). The Euclidean hull of a linear code C is defined in the same fasion and studied in [26].…”

Section: Preliminariesmentioning

confidence: 99%

“…To study constacyclic and QT Hermitian self-dual codes, it suffices to restrict the study to the case where o q 2 (λ)|(q + 1). For λ ∈ {1, −1} (or equivalently, o q 2 (λ) ∈ {1, 2}), λ-constacyclic Hermitian self-dual codes have been studied in [26]. In this paper, we give the characterization and enumeration of λ-constacyclic Hermitian self-dual codes of any length n and over F q 2 for every nonzero λ ∈ F q 2 such that o q 2 (λ)|(q + 1).…”

Section: Introductionmentioning

confidence: 99%

Constacyclic and quasi-twisted Hermitian self-dual codes over finite fields

Sangwisut¹,

Jitman²,

Udomkavanich³

2017

Advances in Mathematics of Communications

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Constacyclic and quasi-twisted Hermitian self-dual codes over finite fields are studied. An algorithm for factorizing x n − λ over F q 2 is given, where λ is a unit in F q 2 . Based on this factorization, the dimensions of the Hermitian hulls of λ-constacyclic codes of length n over F q 2 are determined. The characterization and enumeration of constacyclic Hermitian self-dual (resp., complementary dual) codes of length n over F q 2 are given through their Hermitian hulls. Subsequently, a new family of MDS constacyclic Hermitian self-dual codes over F q 2 is introduced.As a generalization of constacyclic codes, quasi-twisted Hermitian self-dual codes are studied. Using the factorization of x n − λ and the Chinese Remainder Theorem, quasi-twisted codes can be viewed as a product of linear codes of shorter length some over extension fields of F q 2 . Necessary and sufficient conditions for quasi-twisted codes to be Hermitian self-dual are given. The enumeration of such self-dual codes is determined as well.

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“…In [23], this study has been extended to the class of cyclic codes over finite fields and the average hull dimension of cyclic codes has been determined. Later, the hull dimensions of cyclic and negacyclic codes and the number of cyclic codes whose hulls share a given dimension have been established in [17]. The average hull dimension of constacyclic codes over finite fields have been given in [8,9,10].…”

Section: Introductionmentioning

confidence: 99%