Data di Pubblicazione:
2020
Citazione:
HELP: a sparse error locator polynomial for BCH codes / M. Ceria, T. Mora, M. Sala. - In: APPLICABLE ALGEBRA IN ENGINEERING COMMUNICATION AND COMPUTING. - ISSN 0938-1279. - (2020). [Epub ahead of print] [10.1007/s00200-020-00427-x]
Abstract:
In 1990 Cooper suggested to use Groebner bases’ computations to decode
cyclic codes and his idea gave rise to many research papers. In particular, as proved
by Sala-Orsini, once defined the polynomial ring whose variables are the syndromes,
the locations and the error values and considered the syndrome ideal, only one polynomial of a lexicographical Groebner basis for such ideal is necessary to decode (the
general error locator polynomial, a.k.a. GELP). The decoding procedure only consists
in evaluating this polynomial in the syndromes and computing its roots: the roots are
indeed the error locations. A possible bottleneck in this procedure may be the evaluation part, since a priori the GELP may be dense.
In this paper, focusing on binary cyclic codes with length n = 2^m − 1, correcting up to
two errors, we give a Groebner-free, sparse analog of the GELP, the half error locator polynomial (HELP). In particular, we show that it is not necessary to compute the
whole Groebner basis for getting such kind of locator polynomial and we construct
the HELP, studying the quotient algebra of the polynomial ring modulo the syndrome
ideal by a combinatorial point of view. The HELP turns out to be computable with
quadratic complexity and it has linear growth in the length n of the code: O( (n+1)/2) .
Tipologia IRIS:
01 - Articolo su periodico
Keywords:
cyclic codes; syndrome variety; locator polynomial;
Elenco autori:
M. Ceria, T. Mora, M. Sala
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