A generalization of the Berlekamp-Massey algorithm for multisequence shift-register synthesis with applications to decoding cyclic codes

Les codes algébriques principaux et leur décodage

“…On cherche alors des récurrences f = (f s ) s∈N m communes à tous les tableaux, c'est-à-dire telles que Il existe déjà une généralisation de l'algorithme de Berlekamp-Massey pour la construction d'une récurrence linéaire satisfaite simultanément par M suites linéaires [14]. Sakata [38] a construit cette même généralisation de son algorithme, dans le cas de M tableaux m-dimensionels.…”

“…. , n} deux choses l'une : ou bien f (x i ) = y i , ou bien E(x i ) = 0., Ce qui se traduit par (14) f…”

On the Roth and Ruckenstein equations for the Guruswami-Sudan algorithm

“…Based on the former work of Feng and Tzeng [3], [4] the proposed algorithm in [2] for an horizontal band of l Hankel matrices S t has a time complexity of O(lτ 2 ) and a space complexity of O(lτ ). It should be mentioned that for the preprocessing of the syndrome polynomials, an explicit expression of the syndrome coefficients S t i was derived:…”

“…Beside their well-known factorization method Roth and Ruckenstein [2] have also derived a so-called Extended Key Equation (EKE), which is a generalization of the classical Key Equation for unique decoding, by reformulating the interpolation condition of Sudan. The EKE can be solved by an adaption of the Fundamental Iterative Algorithm (FIA) (presented in [3], [4]) for the case of l horizontally arranged Hankel-matrices (where l is the list size). Guruswami and Sudan [5] extended the originally interpolation approach by using multiplicities of higher order.…”

More on solving systems of power equations

“…We divide the system into p − 1 subsystem of linearfeedback shift registers and devise a generalized Berlekamp algorithm to compute the minimum-length characteristic polynomial with only quadratic complexity O(n 2 ). The proposed generalized Berlekamp algorithm is akin to the fundamental iterative algorithm which generalizes the Berlekamp-Massey algorithm to solve multi-sequence linear-feedback shift registers [4]. As an immediate application, the proposed algebraic solution can be used to complement the hard-decision decoding of ternary BCH codes.…”