Nonlinearity and Kernel of Z2^s-Linear Simplex and MacDonald Codes

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Abstract

Z{2^s}-additive codes are subgroups of Z_{2^s}^n , and can be seen as a generalization of linear codes over Z_2 and Z_4. A Z_{2^s}-linear code is a binary code (not necessarily linear) which is the Gray map image of a Z_{2^s}-additive code. We consider Z_{2^s}-additive simplex codes of type α and β, which are a generalization over Z_{2^s} of the binary simplex codes. These codes are related to the Z_{2^s}-additive Hadamard codes. In this paper, we use this relationship to find a linear subcode of the corresponding Z_{2^s}-linear codes, called kernel, and a representation of these codes as cosets of this kernel. In particular, this also gives the linearity of these codes. Similarly, Z_{2^s}-additive MacDonald codes are defined for s > 2, and equivalent results are obtained.
Original languageEnglish
Article number11
Pages (from-to)7174-7183
Number of pages10
JournalIEEE Transactions on Information Theory
Volume68
Issue number11
DOIs
Publication statusPublished - Nov 2022

Keywords

  • Gray map
  • Hadamard codes
  • MacDonald codes
  • cosets
  • simplex codes
  • ℤ -linear codes

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