On Self-Dual Four Circulant Codes

Shi, Minjia; Hong-wei, Zhu; Qian, Liqin; Solé, Patrick

doi:10.1142/s0129054118500259

Cited by 14 publications

(6 citation statements)

References 13 publications

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“…On the other side, double circulant LCD codes over Z 4 in [22], Z p 2 in [10] and Galois ring in [23] were studied, respectively. Further, for some related studies on these topics, interested readers can see [20,24,26,27,29]. Therefore, because of available works on non-chain rings [21,28], it is logical to investigate these codes over other semi-local non-chain rings.…”

Section: Introductionmentioning

confidence: 99%

Self-dual and LCD double circulant and double negacirculant codes over $${\mathbb {F}}_q+u{\mathbb {F}}_q+v{\mathbb {F}}_q$$

Yadav

Islam

Prakash

et al. 2021

J. Appl. Math. Comput.

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Let q be an odd prime power, and denote by F q the finite field with q elements. In this paper, we consider the ring R = F q + uF q + vF q , where u 2 = u, v 2 = v, uv = vu = 0 and study double circulant and double negacirculant codes over this ring. We first obtain the necessary and sufficient conditions for a double circulant code to be self-dual (resp. LCD). Then we enumerate self-dual and LCD double circulant and double negacirculant codes over R. Last but not the least, we show that the family of Gray images of self-dual and LCD double circulant codes over R are good. Several numerical examples of self-dual and LCD codes over F 5 as the Gray images of these codes over R are given in short lengths.

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Section: Introductionmentioning

confidence: 99%

Self-dual and LCD double circulant and double negacirculant codes over $${\mathbb {F}}_q+u{\mathbb {F}}_q+v{\mathbb {F}}_q$$

Yadav

Islam

Prakash

et al. 2021

J. Appl. Math. Comput.

Self Cite

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“…Later, circulant matrices have been shown to have applications in many disciplines, e.g., signal processing, image processing, networked systems, communications, and coding theory. Especially, (nonsingular) circulant matrices over nite elds and over commutative nite chain rings are applied in constructions of various families of linear codes (see [1], [2], [6], [9], [10], [11], [14], [17], [18], [19], and references therein). Circulant matrices have shown to have a closed connection with diagonal matrices (see, for example, [6] and [14]).…”

Section: Introductionmentioning

confidence: 99%

Determinants of some special matrices over commutative finite chain rings

Jitman

2020

Special Matrices

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Circulant matrices over finite fields and over commutative finite chain rings have been of interest due to their nice algebraic structures and wide applications. In many cases, such matrices over rings have a closed connection with diagonal matrices over their extension rings. In this paper, the determinants of diagonal and circulant matrices over commutative finite chain rings R with residue field 𝔽q are studied. The number of n × n diagonal matrices over R of determinant a is determined for all elements a in R and for all positive integers n. Subsequently, the enumeration of nonsingular n × n circulant matrices over R of determinant a is given for all units a in R and all positive integers n such that gcd(n, q) = 1. In some cases, the number of singular n × n circulant matrices over R with a fixed determinant is determined through the link between the rings of circulant matrices and diagonal matrices. As applications, a brief discussion on the determinants of diagonal and circulant matrices over commutative finite principal ideal rings is given. Finally, some open problems and conjectures are posted

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“…Later, circulant matrices have been shown to have applications in many disciplines, e.g., signal processing, image processing, networked systems, communications, and coding theory. Especially, (nonsingular) circulant matrices over finite fields and over commutative finite chain rings are applied in constructions of various families of linear codes (see [1], [5], [7], [9], [10], [11], [14], [17], [18], [19] , and references therein). Circulant matrices have shown to have a closed connection with diagonal matrices (see, for example, [5] and [14]).…”

Section: Introductionmentioning

confidence: 99%

Determinants of some Special Matrices over Commutative Finite Chain Rings

Jitman¹

2020

Preprint

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Circulant matrices over finite fields and over commutative finite chain rings have been of interest due to their nice algebraic structures and wide applications. In many cases, such matrices over rings have a closed connection with diagonal matrices over their extension rings. In this paper, the determinants of diagonal and circulant matrices over commutative finite chain rings R with residue field F q are studied. The number of n × n diagonal matrices over R of determinant a is determined for all elements a in R and for all positive integers n. Subsequently, the enumeration of nonsingular n × n circulant matrices over R of determinant a is given for all units a in R and all positive integers n such that gcd(n, q) = 1. In some cases, the number of singular n × n circulant matrices over R with a fixed determinant is determined through the link between the rings of circulant matrices and diagonal matrices. As applications, a brief discussion on the determinant of diagonal and circulant matrices over commutative finite principal ideal rings is given. Finally, some open problems

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On Self-Dual Four Circulant Codes

Cited by 14 publications

References 13 publications

Self-dual and LCD double circulant and double negacirculant codes over $${\mathbb {F}}_q+u{\mathbb {F}}_q+v{\mathbb {F}}_q$$

Self-dual and LCD double circulant and double negacirculant codes over $${\mathbb {F}}_q+u{\mathbb {F}}_q+v{\mathbb {F}}_q$$

Determinants of some special matrices over commutative finite chain rings

Determinants of some Special Matrices over Commutative Finite Chain Rings

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