On Z2Z4-additive polycyclic codes and their Gray images View Full Text


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Article Info

DATE

2021-08-27

AUTHORS

Rongsheng Wu, Minjia Shi

ABSTRACT

In this paper, we first generalize the polycyclic codes over finite fields to polycyclic codes over the mixed alphabet Z2Z4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_2\mathbb {Z}_4$$\end{document}, and we show that these codes can be identified as Z4[x]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_4[x]$$\end{document}-submodules of Rα,β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_{\alpha ,\beta }$$\end{document} with Rα,β=Z2[x]/⟨t1(x)⟩×Z4[x]/⟨t2(x)⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_{\alpha ,\beta }=\mathbb {Z}_2[x]/\langle t_1(x)\rangle \times \mathbb {Z}_4[x]/\langle t_2(x)\rangle $$\end{document}, where t1(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_1(x)$$\end{document} and t2(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_2(x)$$\end{document} are monic polynomials over Z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_2$$\end{document} and Z4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_4$$\end{document}, respectively. Then we provide the generator polynomials and minimal generating sets for this family of codes based on the strong Gröbner basis. In particular, under the proper defined inner product, we study the dual of Z2Z4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_2\mathbb {Z}_4$$\end{document}-additive polycyclic codes. Finally, we focus on the characterization of the Z2Z4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_2\mathbb {Z}_4$$\end{document}-MDSS and MDSR codes, and as examples, we also present some (almost) optimal binary codes derived from the Z2Z4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_2\mathbb {Z}_4$$\end{document}-additive polycyclic codes. More... »

PAGES

1-12

References to SciGraph publications

  • 2020-11-24. ZpZp[v]-additive cyclic codes are asymptotically good in JOURNAL OF APPLIED MATHEMATICS AND COMPUTING
  • 2017-08-04. A characterization of Z2Z2[u]-linear codes in DESIGNS, CODES AND CRYPTOGRAPHY
  • 2009-08-01. -linear codes: generator matrices and duality in DESIGNS, CODES AND CRYPTOGRAPHY
  • 2010-10-05. Maximum distance separable codes over and in DESIGNS, CODES AND CRYPTOGRAPHY
  • 2019-08-14. One-weight and two-weight ℤ2ℤ2[u,v]-additive codes in CRYPTOGRAPHY AND COMMUNICATIONS
  • 2019-08-23. ℤpℤps-additive cyclic codes are asymptotically good in CRYPTOGRAPHY AND COMMUNICATIONS
  • 2015-09-23. One weight Z2Z4 additive codes in APPLICABLE ALGEBRA IN ENGINEERING, COMMUNICATION AND COMPUTING
  • 2007-08-11. On repeated-root multivariable codes over a finite chain ring in DESIGNS, CODES AND CRYPTOGRAPHY
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    http://scigraph.springernature.com/pub.10.1007/s10623-021-00917-0

    DOI

    http://dx.doi.org/10.1007/s10623-021-00917-0

    DIMENSIONS

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    30 binary codes
    31 characterization
    32 code
    33 example
    34 family
    35 family of codes
    36 field
    37 finite field
    38 generating set
    39 generator polynomials
    40 gray image
    41 images
    42 inner product
    43 minimal generating set
    44 mixed alphabets
    45 monic polynomials
    46 optimal binary codes
    47 paper
    48 polycyclic codes
    49 polynomials
    50 products
    51 set
    52 strong Gröbner bases
    53 submodules
    54 submodules of Rα
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