21792 articles – 15600 references  [version française]
 HAL: inria-00509219, version 1
 Journal of Symbolic Computation 44, 12 (2009) 1608-1625
 On the decoding of binary cyclic codes with the Newton's identities
 (2009-12-31)
 We revisit in this paper the concept of decoding binary cyclic codes with Gröbner bases. These ideas were first introduced by Cooper, then Chen, Reed, Helleseth and Truong, and eventually by Orsini and Sala. We discuss here another way of putting the decoding problem into equations: the Newton's identities. Although these identities have been extensively used for decoding, the work was done manually, to provide formulas for the coefficients of the locator polynomial. This was achieved by Reed, Chen, Truong and others in a long series of papers, for decoding quadratic residue codes, on a case-by-case basis. It is tempting to automate these computations, using elimination theory and Gröbner bases. Thus, we study in this paper the properties of the system defined by the Newton's identities, for decoding binary cyclic codes. This is done in two steps, first we prove some facts about the variety associated to this system, then we prove that the ideal itself contains relevant equations for decoding, which lead to formulas. Then we consider the so-called online Gröbner bases decoding, where the work of computing a Gröbner basis is done for each received word. It is much more efficient for practical purposes than preprocessing and substituting into the formulas. Finally, we conclude with some computational results, for codes of interesting length (about one hundred).
 1: SECRET (INRIA Rocquencourt) INRIA 2: SALSA (INRIA Rocquencourt) INRIA – CNRS : UMR7606 – Université Pierre et Marie Curie [UPMC] - Paris VI 3: Laboratoire d'Informatique, de Traitement de l'Information et des Systèmes (LITIS) Institut National des Sciences Appliquées (INSA) - Rouen – Université du Havre – Université de Rouen : EA4108
 Domain : Computer Science/Symbolic ComputationComputer Science/Information Theory and CodingMathematics/Information Theory
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 inria-00509219, version 1 http://hal.inria.fr/inria-00509219 oai:hal.inria.fr:inria-00509219 From: Daniel Augot <> Submitted on: Tuesday, 10 August 2010 18:27:09 Updated on: Thursday, 12 August 2010 11:57:56