This paper gives concatenated codes with uniform construction complexity of order n

2 Dierent kinds of cyclic codes. 4 2.1 Notation.............................. 5 2.2 Denitions............................. 6

Abstract not found

We address the problem of the algebraic decoding of any cyclic code up to the true minimum distance. For this, we use the classical formulation of the problem, which is to find the error locator polynomial in terms of the syndromes of the received word. This is usually done with the Berlekamp-Massey algorithm in the case of BCH codes and related codes, but for the general case, there is no generic algorithm to decode cyclic codes. Even in the case of the quadratic residue codes, which are good codes with a very strong algebraic structure, there is no available general decoding algorithm. For this particular case of quadratic residue codes, several authors have worked out, by hand, formulas for the coefficients of the locator polynomial in terms of the syndromes, using the Newton identities. This work has to be done for each particular quadratic residue code, and is more and more difficult as the length is growing. Furthermore, it is error-prone. We propose to automate these computations, using elimination theory and Gröbner bases. We prove that, by computing appropriate Gröbner bases, one automatically recovers formulas for the coefficients of the locator polynomial, in terms of the syndromes.

This paper consider the use of Newton’s identities for establishing properties of cyclic codes. The main tool is to consider these identities as equations, and to look for the properties of the solutions. First these equations have been considered as necessary conditions for establishing non existence properties of cyclic codes, such as the non existence of codewords of a given weight. The properties of these equations are studied, and the properties of the solution to the algebraic system are given. The main theorem is that codewords in a hamming sphere around a given word can be characterized by algebraic conditions. This theorem enables to describe the minimum codewords of a given cyclic codes, by algebraic conditions. The equations are solved using the Buchberger’s algorithm for computing a Groebner basis. Examples are also given with alternant codes, and with a non linear code.