Families of hadamard Z<inf>2Z4</inf>Z<inf>8</inf>-codes

Angel Del Rio, Josep Rifa

Research output: Contribution to journalArticleResearchpeer-review

4 Citations (Scopus)

Abstract

A Z2Z4Q8-code is the binary image, after a Gray map, of a subgroup of BBZ2k-1×\BBZ4k 2× Q8 k3, where Q8 is the quaternion group on eight elements. Such BBZ2\BBZ4Q8-codes are translation invariant propelinear codes as are the well known BBZ 4-linear or BBZ2BBZ4-linear codes. In this paper, we show that there exist 'pure' BBZ-2 BBZ4Q 8-codes, that is, codes that do not admit any abelian translation invariant propelinear structure. We study the dimension of the kernel and rank of the BBZ 2\BBZ4Q8-codes, and we give upper and lower bounds for these parameters. We give tools to construct a new class of Hadamard codes formed by several families of BBZ2\BBZ 4Q8-codes; we classify such codes from an algebraic point of view and we improve the upper and lower bounds for the rank and the dimension of the kernel when the codes are Hadamard. © 2013 IEEE.
Original languageEnglish
Article number6508950
Pages (from-to)5140-5151
JournalIEEE Transactions on Information Theory
Volume59
Issue number8
DOIs
Publication statusPublished - 29 Jul 2013

Keywords

  • 1-perfect codes
  • BBZ BBZ Q -codes 2 4 8
  • Hadamard codes
  • propelinear codes
  • translation invariant codes

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