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Étude algèbrique des mots de poids minimum des codes cycliques, méthodes d'algèbre linéaire sur les corps finis.

Daniel Augot 1
1 CODES - Coding and cryptography
Inria Paris-Rocquencourt
Abstract : Minimum weight codewords of cyclic error-correcting codes are considered here. The elementary symmetric functions and the power-sum symmetric functions of the locators of these codewords are related by the Newton's identities. In the first chapter, they are viewed as a system of algebraic equations, whose solutions are studied by means of the Fourier transform. In chapter II, the link with cyclic error-correcting codes is established. On some examples, it is shown how to study the minimum weight codewords on the data of a Grobner basis of the ideal generated by the Newton's identities. In chapter III, the Newton's identities are considered from a theoretical point of view, and results about minimum weight codewords of a family of BCH codes are obtained. These computations occurs in the context of finite fields. In chapter IV, an algorithm is constructed for computing a normal basis of a finite field. A point of view from linear algebra is chosen, and other problems are dealt with (the computation of the minimal polynomial, of the Frobenius form of a matrix, when the factorization of the characteristic polynomial is known).
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Daniel Augot. Étude algèbrique des mots de poids minimum des codes cycliques, méthodes d'algèbre linéaire sur les corps finis.. Théorie de l'information [cs.IT]. Université Pierre et Marie Curie - Paris VI, 1993. Français. ⟨tel-00723227⟩

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