On self-dual negacirculant codes of index two and four View Full Text


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Article Info

DATE

2018-01-22

AUTHORS

Minjia Shi, Liqin Qian, Patrick Solé

ABSTRACT

We study the asymptotic performance of quasi-twisted codes viewed as modules in the ring R=Fq[x]/⟨xn+1⟩,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R=\mathbb {F}_q[x]/\langle x^n+1\rangle , $$\end{document} when they are self-dual and of length 2n or 4n. In particular, in order for the decomposition to be amenable to analysis, we study factorizations of xn+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x^n+1$$\end{document} over Fq,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_q, $$\end{document} with n twice an odd prime, containing only three irreducible factors, all self-reciprocal. We give arithmetic conditions bearing on n and q for this to happen. Given a fixed q, we show these conditions are met for infinitely many n’s, provided a refinement of Artin primitive root conjecture holds. This number theory conjecture is known to hold under generalized Riemann hypothesis (GRH). We derive a modified Varshamov–Gilbert bound on the relative distance of the codes considered, building on exact enumeration results for given n and q. More... »

PAGES

2485-2494

References to SciGraph publications

  • 2007-08. Extremal Ternary Self-Dual Codes Constructed from Negacirculant Matrices in GRAPHS AND COMBINATORICS
  • 1993-06. Explicit factorization of + 1 overFp with primep≡3 mod 4 in APPLICABLE ALGEBRA IN ENGINEERING, COMMUNICATION AND COMPUTING
  • 2017-08-19. Good integers and some applications in coding theory in CRYPTOGRAPHY AND COMMUNICATIONS
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    http://scigraph.springernature.com/pub.10.1007/s10623-017-0455-0

    DOI

    http://dx.doi.org/10.1007/s10623-017-0455-0

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