On Z 2s -linear Hadamard codes: kernel and partial classification

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The Z2s-additive codes are subgroups of Z2sn, and can be seen as a generalization of linear codes over Z 2 and Z 4 . A Z2s-linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z2s-additive code. It is known that the dimension of the kernel can be used to give a complete classification of the Z 4 -linear Hadamard codes. In this paper, the kernel of Z2s-linear Hadamard codes of length 2 t and its dimension are established for s> 2. Moreover, we prove that this invariant only provides a complete classification for some values of t and s. The exact amount of nonequivalent such codes are given up to t= 11 for any s≥ 2 , by using also the rank.
Original languageEnglish
Pages (from-to)417-435
Number of pages19
JournalDesigns, Codes, and Cryptography
Issue number2-3
Publication statusPublished - 15 Mar 2019


  • Classification
  • Gray map
  • Hadamard code
  • Kernel
  • Z2s-additive code
  • Z2s-linear code


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