Algebraic decoding of negacyclic codes over $Bbb Z_4$

Eimear Byrne, Marcus Greferath, Jaume Pernas, Jens Zumbrägel

Research output: Contribution to journalArticleScientificpeer-review

5 Citations (Scopus)

Abstract

In this article we investigate Berlekamp’s negacyclic codes and discover that these codes, when considered over the integers modulo 4, do not suffer any of the restrictions on the minimum distance observed in Berlekamp’s original papers: our codes have minimum Lee distance at least 2t + 1, where the generator polynomial of the code has roots α, α3, . . . , α2t-1 for a primitive 2nth root α of unity in a Galois extension of Z4 ; no restriction on t is imposed. We present an algebraic decoding algorithm for this class of codes that corrects any error pattern of Lee weight ≤ t. Our treatment uses Gröbner bases, the decoding complexity is quadratic in t.
Original languageUndefined/Unknown
Pages (from-to)3-16
Number of pages14
JournalDESIGNS CODES AND CRYPTOGRAPHY
Volume66
Issue number1-3
DOIs
Publication statusPublished - 2013
MoE publication typeA1 Journal article-refereed

Keywords

  • negacyclic code
  • integers modulo 4
  • lee metric
  • Galois ring
  • decoding
  • Gröbner bases
  • ey equation
  • solution by approximations
  • module of solutions

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