ℤ<inf>2</inf>ℤ<inf>4</inf>-linear codes: Rank and kernel

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Abstract

A code ℤ2ℤ4-additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C by deleting the coordinates outside X (respectively, Y) is a binary linear code (respectively, a quaternary linear code). The corresponding binary codes of ℤ2ℤ4-additive codes under an extended Gray map are called ℤ2ℤ4-linear codes. In this paper, the invariants for ℤ2ℤ4-linear codes, the rank and dimension of the kernel, are studied. Specifically, given the algebraic parameters of ℤ2ℤ4-linear codes, the possible values of these two invariants, giving lower and upper bounds, are established. For each possible rank r between these bounds, the construction of a ℤ2ℤ4-linear code with rank r is given. Equivalently, for each possible dimension of the kernel k, the construction of a ℤ2ℤ4-linear code with dimension of the kernel k is given. Finally, the bounds on the rank, once the kernel dimension is fixed, are established and the construction of a ℤ2ℤ4- linear code for each possible pair (r, k) is given. © 2009 Springer Science+Business Media, LLC.
Original languageEnglish
Pages (from-to)43-59
JournalDesigns, Codes, and Cryptography
Volume56
DOIs
Publication statusPublished - 1 Jul 2010

Keywords

  • Kernel
  • Quaternary linear codes
  • Rank
  • ℤ -linear codes 4
  • ℤ ℤ -linear codes 2 4
  • ℤ ℤ -additive codes 2 4

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