Recurrent Sequences in Finite Fields and Cyclic Linear Codes

Self-dual cyclic codes form an important class of linear codes. It has been shown that there exists a self-dual cyclic code of length [Formula: see text] over a finite field if and only if [Formula: see text] and the field characteristic are even. The enumeration of such codes has been given under both the Euclidean and Hermitian products. However, in each case, the formula for self-dual cyclic codes of length [Formula: see text] over a finite field contains a characteristic function which is not easily computed. In this paper, we focus on more efficient ways to enumerate self-dual cyclic codes of lengths [Formula: see text] and [Formula: see text], where [Formula: see text], [Formula: see text], and [Formula: see text] are positive integers. Some number theoretical tools are established. Based on these results, alternative formulas and efficient algorithms to determine the number of self-dual cyclic codes of such lengths are provided.

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On Group Codes Over Elementary Abelian Groups

Sultan Qaboos University Journal for Science [SQUJS] ◽

10.24200/squjs.vol8iss2pp145-151 ◽

2003 ◽

Vol 8 (2) ◽

pp. 145

Author(s):

Adnan A. Zain

Keyword(s):

Abelian Group ◽

Finite Fields ◽

Linear Codes ◽

Abelian Groups ◽

Parity Check ◽

Group Codes ◽

New Codes ◽

Generator Matrices

For group codes over elementary Abelian groups we present definitions of the generator and the parity check matrices, which are matrices over the ring of endomorphism of the group. We also lift the theorem that relates the parity check and the generator matrices of linear codes over finite fields to group codes over elementary Abelian groups. Some new codes that are MDS, self-dual, and cyclic over the Abelian group with four elements are given.

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Constructions of (r,t)-LRC Based on Totally Isotropic Subspaces in Symplectic Space Over Finite Fields

International Journal of Foundations of Computer Science ◽

10.1142/s0129054120500112 ◽

2020 ◽

Vol 31 (03) ◽

pp. 327-339

Author(s):

Gang Wang ◽

Min-Yao Niu ◽

Fang-Wei Fu

Keyword(s):

Finite Fields ◽

Linear Codes ◽

Distributed Storage ◽

Symplectic Space ◽

Information Rate ◽

Local Repair ◽

Distributed Storage Systems ◽

Isotropic Subspaces ◽

Locally Repairable Codes ◽

Totally Isotropic Subspaces

Linear code with locality [Formula: see text] and availability [Formula: see text] is that the value at each coordinate [Formula: see text] can be recovered from [Formula: see text] disjoint repairable sets each containing at most [Formula: see text] other coordinates. This property is particularly useful for codes in distributed storage systems because it permits local repair of failed nodes and parallel access of hot data. In this paper, two constructions of [Formula: see text]-locally repairable linear codes based on totally isotropic subspaces in symplectic space [Formula: see text] over finite fields [Formula: see text] are presented. Meanwhile, comparisons are made among the [Formula: see text]-locally repairable codes we construct, the direct product code in Refs. [8], [11] and the codes in Ref. [9] about the information rate [Formula: see text] and relative distance [Formula: see text].

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