The second weight of the Generalized Reed-Muller code of length q n and order d over the finite field with q elements is now known for d < q and d > (n − 1)(q − 1). In this paper, we determine the second weight for the other values of d which are not multiples of q − 1 plus 1. For the special case d = a(q − 1) + 1 we give an estimate.
We propose new results on low weight codewords of affine and projective generalized Reed-Muller codes. In the affine case we prove that if the size of the working finite field is large compared to the degree of the code, the low weight codewords are products of affine functions. Then in the general case we study some types of codewords and prove that they… (More)
— Let us consider an algebraic function field defined over a finite Galois extension K of a perfect field k. We recall some elementary conditions allowing the descent of the definition field of the algebraic function field from K to k. We apply these results to the descent of the definition field of a tower of function fields. We give explicitly the… (More)
— The aim of this paper is to introduce the bilinear complexity of the multiplication in finite fields and to give a brief exposition of the recent results obtained in this part of algebraic complexity theory. In particular we present the new results obtained using the Chudnovsky-Chudnovsky algorithm and its generalizations. Résumé (Sur la complexité… (More)
A brief survey on low weight codewords of generalized Reed-Muller codes and projective generalized Reed-Muller codes is presented. In the affine case some information about the words that reach the second distance is given. Moreover the second weight of the projective Reed-Muller codes is estimated, namely a lower bound and an upper bound of this weight are… (More)
This paper focuses on parallel hash functions based on tree modes of operation for a compression function. We discuss the various forms of opti-mality that can be obtained when designing such parallel hash functions. The first result is a scheme which optimizes the tree topology in order to decrease at best the running time. Then, without affecting the… (More)
Soit q = p s une puissance d'un nombre premier p. Nous noterons F q le corps finì a q ´ eléments.
We study the geometrical properties of the subgroups of the mutliplicative group of a "nite extension of a "nite "eld endowed with its vector space structure and we show that in some cases the associated projective space has a natural group structure. We construct some cyclic codes related to Reed}Muller codes by evaluating polynomials on these subgroups.… (More)
In this paper we propose a signature scheme based on two intractable problems , namely the integer factorization problem and the discrete logarithm problem for elliptic curves. It is suitable for applications requiring long-term security and provides smaller signatures than the existing schemes based on the integer factor-ization and integer discrete… (More)