ZpZp2-linear codes: rank and kernel View Full Text


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Article Info

DATE

2021-10-29

AUTHORS

Minjia Shi, Shukai Wang, Xiaoxiao Li

ABSTRACT

A code C is called ZpZp2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_p\mathbb {Z}_{p^2}$$\end{document}-linear if it is the Gray image of a ZpZp2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_p\mathbb {Z}_{p^2}$$\end{document}-additive code, where p>2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>2$$\end{document} is prime. In this paper, the rank and the dimension of the kernel of ZpZp2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_p\mathbb {Z}_{p^2}$$\end{document}-linear codes are studied. The range of their values is given. For each value of the rank of ZpZp2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_p\mathbb {Z}_{p^2}$$\end{document}-linear codes, we give detailed construction of the corresponding codes. Similarly, for the values of the dimension of the kernel, we also give a construction method. Finally, pairs of rank and the dimension of the kernel of ZpZp2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_p\mathbb {Z}_{p^2}$$\end{document}-additive codes are also considered. More... »

PAGES

1-14

References to SciGraph publications

  • 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
  • 2014-03-04. Permutation decoding of Z2Z4-linear codes 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
  • 2016-12-31. There is exactly one Z2Z4-cyclic 1-perfect code in DESIGNS, CODES AND CRYPTOGRAPHY
  • 2019-08-23. ℤpℤps-additive cyclic codes are asymptotically good in CRYPTOGRAPHY AND COMMUNICATIONS
  • 2008-01-01. On Rank and Kernel of ℤ4-Linear Codes in CODING THEORY AND APPLICATIONS
  • 2015-09-23. One weight Z2Z4 additive codes in APPLICABLE ALGEBRA IN ENGINEERING, COMMUNICATION AND COMPUTING
  • 2009-11-06. -linear codes: rank and kernel in DESIGNS, CODES AND CRYPTOGRAPHY
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