Equivalences among Z2s -linear Hadamard codes

Cristina Fernández-Córdoba*, Carlos Vela, Mercè Villanueva

*Corresponding author for this work

Research output: Contribution to journalArticleResearchpeer-review

1 Citation (Scopus)

Abstract

The Z2s -additive codes are subgroups of Z2s n, and can be seen as a generalization of linear codes over Z2 and Z4. A Z2s -linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z2s -additive code. A partial classification of these codes by using the dimension of the kernel is known. In this paper, we establish that some Z2s -linear Hadamard codes of length 2t are equivalent, once t is fixed. This allows us to improve the known upper bounds for the number of such nonequivalent codes. Moreover, up to t=11, this new upper bound coincides with a known lower bound (based on the rank and dimension of the kernel). Finally, when we focus on s∈{2,3}, the full classification of the Z2s -linear Hadamard codes of length 2t is established by giving the exact number of such codes.

Original languageEnglish
Article number111721
JournalDiscrete Mathematics
Volume343
Issue number3
DOIs
Publication statusPublished - Mar 2020

Keywords

  • Classification
  • Gray map
  • Hadamard code
  • Kernel
  • Rank
  • Z-additive code

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