VARIOUS STRUCTURES OF CYCLIC CODES OVER THE NON-FROBENIUS RING Fp[u, v]/>au2, v2, uv, vu5b

被引：0

作者：

Choi, Hyun Seung ^{[1
]}

Kim, Boran ^{[1
]}

机构：

[1] Kyungpook Natl Univ, Dept Math Educ, Daegu 41566, South Korea

来源：

ADVANCES IN MATHEMATICS OF COMMUNICATIONS | 2024年 / 18卷 / 02期

基金：

新加坡国家研究基金会;

关键词：

Cyclic codes; Self-orthogonal codes; Self-dual codes; LCD codes; Mass formulae; NEGACYCLIC CODES; LCD CODES; ENUMERATION; CLASSIFICATION;

D O I：

10.3934/amc.2023030

中图分类号：

TP301 [理论、方法];

学科分类号：

081202 ;

摘要：

. In this paper, we study cyclic codes over the ring Rp = Fp+uFp+ vFp, where u2 = v2 = uv = vu = 0 (p: a prime number). We first derive generators of ideals of Rp[x]/(xn - 1), which are corresponding to cyclic codes over Rp of arbitrary length n. Especially, for gcd(n, p) = 1, we have the explicit generators of ideals corresponding to cyclic codes, their duals, self-orthogonal codes and self-dual codes over Rp of length n. Furthermore, mass formulae of cyclic self-orthogonal and LCD codes over Rp of length n is obtained. Finally, we present a Gray map from (Rp)n to (Fp)6n, showing that the image of a cyclic code over this Gray map is quasi-cyclic, and determine the index of this image. A series of examples and tables concerning the mass formula of cyclic self-orthogonal codes, cyclic LCD codes and several new quasi-cyclic codes are presented as an application of theorems.

引用

页码：304 / 327

页数：24

共 49 条

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[34] Quantum Codes from Cyclic Codes over the Ring R = IFq + uIFq + vIFq + uvIFq + v2IFq + uv2IFq
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[35] Constacyclic codes over the ring Fp[u,v]/⟨u2-1,v3-v,uv-vu⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_p[u, v]/\langle u^2-1, v^3-v, uv-vu\rangle $$\end{document} and their applications
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[36] Cyclic DNA codes over the ring F2 + uF2 + vF2 + uvF2 + v2F2 + uv2F2
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[37] Quantum codes from the cyclic codes over Fp[u,v,w]/⟨u2-1,v2-1,w2-1,uv-vu,vw-wv,wu-uw⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_{p}[u,v,w]/\langle u^{2}-1,v^{2}-1,w^{2}-1,uv-vu,vw-wv,wu-uw\rangle $$\end{document}
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[38] Skew constacyclic codes over a non-chain ring Fq[u,v]/⟨f(u),g(v),uv-vu⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {F}}_{q}[u,v]/\langle f(u),g(v), uv-vu\rangle $$\end{document}
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[39] Polyadic constacyclic codes over a non-chain ring Fq[u,v]/⟨f(u),g(v),uv-vu⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_{q}[u,v]/\langle f(u),g(v), uv-vu\rangle $$\end{document}
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[40] (θ,δθ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\theta , \delta _\theta )$$\end{document}-Cyclic codes over Fq[u,v]/⟨u2-u,v2-v,uv-vu⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_q[u,v]/\langle u^2-u, v^2-v, uv-vu \rangle $$\end{document}
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VARIOUS STRUCTURES OF CYCLIC CODES OVER THE NON-FROBENIUS RING Fp[u, v]/&gt;au2, v2, uv, vu5b

VARIOUS STRUCTURES OF CYCLIC CODES OVER THE NON-FROBENIUS RING Fp[u, v]/>au2, v2, uv, vu5b