New classes of binary few weight codes from trace codes over a chain ring View Full Text


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Article Info

DATE

2021-07-19

AUTHORS

Xiaoxiao Li, Minjia Shi

ABSTRACT

In this paper, we construct several infinite families of codes over the chain ring R=F2[u]/⟨uk⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R={\mathbb {F}}_2[u]/\langle u^k\rangle $$\end{document}, i.e., R=F2+uF2+⋯+uk-1F2,uk=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R={\mathbb {F}}_2+u{\mathbb {F}}_2+\cdots +u^{k-1}{\mathbb {F}}_2, u^{k}=0$$\end{document} by employing simplicial complexes. When the simplicial complexes are all generated by a single maximal element, we compute the homogeneous weight distributions of these classes of codes. By the Gray map, we obtain that some classes of codes are minimal and some classes of these codes are distance optimal. The codewords of these codes are shown to be minimal for inclusion of supports, a fact favorable to an application to secret sharing schemes. More... »

PAGES

1-12

References to SciGraph publications

  • 1998-11. How to Build Robust Shared Control Systems in DESIGNS, CODES AND CRYPTOGRAPHY
  • 2016-09-27. Two and three weight codes over Fp+uFp in CRYPTOGRAPHY AND COMMUNICATIONS
  • 2020-09-11. Optimal minimal linear codes from posets in DESIGNS, CODES AND CRYPTOGRAPHY
  • 2017-10-28. New Classes of p-Ary Few Weight Codes in BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY
  • 2018-01-11. Several new classes of linear codes with few weights in CRYPTOGRAPHY AND COMMUNICATIONS
  • 2017-11-24. Linear codes from simplicial complexes in DESIGNS, CODES AND CRYPTOGRAPHY
  • 2003-06-18. Covering and Secret Sharing with Linear Codes in DISCRETE MATHEMATICS AND THEORETICAL COMPUTER SCIENCE
  • 2017-10-24. Few-weight codes from trace codes over a local ring in APPLICABLE ALGEBRA IN ENGINEERING, COMMUNICATION AND COMPUTING
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    http://scigraph.springernature.com/pub.10.1007/s12190-021-01549-2

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    http://dx.doi.org/10.1007/s12190-021-01549-2

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