Few-weight codes from trace codes over a local ring

Ontology type: schema:ScholarlyArticle

Article Info

DATE

2017-10-24

AUTHORS ABSTRACT

In this paper, few weights linear codes over the local ring R=Fp+uFp+vFp+uvFp,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R={\mathbb {F}}_p+u{\mathbb {F}}_p+v{\mathbb {F}}_p+uv{\mathbb {F}}_p,$$\end{document} with u2=v2=0,uv=vu,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u^2=v^2=0, uv=vu,$$\end{document} are constructed by using the trace function defined on an extension ring of degree m of R. These trace codes have the algebraic structure of abelian codes. Their weight distributions are evaluated explicitly by means of Gauss sums over finite fields. Two different defining sets are explored. Using a linear Gray map from R to Fp4,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {F}}_p^4,$$\end{document} we obtain several families of p-ary codes from trace codes of dimension 4m. For two different defining sets: when m is even, or m is odd and p≡3(mod4).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\equiv 3 ~(\mathrm{mod} ~4).$$\end{document} Thus we obtain two family of p-ary abelian two-weight codes, which are directly related to MacDonald codes. When m is even and under some special conditions, we obtain two classes of three-weight codes. In addition, we give the minimum distance of the dual code. Finally, applications of the p-ary image codes in secret sharing schemes are presented. More... »

PAGES

335-350

References to SciGraph publications

• 2008-02-28. Ring geometries, two-weight codes, and strongly regular graphs in DESIGNS, CODES AND CRYPTOGRAPHY
• 2016-09-27. Two and three weight codes over Fp+uFp in CRYPTOGRAPHY AND COMMUNICATIONS
• 2003-06-18. Covering and Secret Sharing with Linear Codes in DISCRETE MATHEMATICS AND THEORETICAL COMPUTER SCIENCE

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http://scigraph.springernature.com/pub.10.1007/s00200-017-0345-8

DOI

http://dx.doi.org/10.1007/s00200-017-0345-8

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