HELP: a sparse error locator polynomial for BCH codes

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 . Keywords Cyclic codes • Syndrome variety • Locator polynomial Mathematics Subject Classification (2010) 94B35 • 94B15 • 05E40 • 13P10 This research has been partially funded by INdAM -

Macaulay, Lazard and the Syndrome Variety

“…Experiments showed that in that setting HELP has a very sparse formula, which has been proved in [12]:…”

“…The shape of N 2 for a [2 m − 1, 2]-code C over F 2 m has been given recently by [12] which moreover proved that the related Gröbner basis has the shape G = (x n 1 − 1, g 2 , z 2 + z 1 + x 1 , g 4 ) where (see [37])…”

“…On the basis of that, an improvement of GELP was introduced, under the label of half error locator polynomial (HELP), in [12], which remarks that the relation…”

“…1 − t∈N c t t is Sala-Orsini general error locator polynomial. Such result allowed [12] to remark (applying Marinari-Mora Theorem [30,1,7]) that, for decoding, it is sufficient to express Z as…”