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The q-ary Reed-Muller codes RM q (u, m) of length n = q m are a generalization of Reed-Solomon codes, which use polynomials in m variables to encode messages through functional encoding. Using an idea of reducing the multivariate case to the uni-variate case, randomized list-decoding algorithms for Reed-Muller codes were given in [1] and [15]. The algorithm… (More)

The concept of an error-correcting array gives a new bound on the minimum distance of linear codes and a decoding algorithm which decodes up to half this bound. This gives a unified point of view which explains several improvements on the minimum distance of algebraic-geometric codes. Moreover it is explained in terms of linear algebra and the theory of… (More)

We generalize the existing decoding algorithms by error location for BCH and algebraic-geometric codes to arbitrary linear codes. We investigate the number of dependent sets of error positions. A received word with an independent set of error positions can be corrected.

The order bound on generalized Hamming weights is introduced in a general setting of codes on varieties which comprises both the one point geometric Goppa codes as the q-ary Reed-Muller codes. For the latter codes it is shown that this bound is sharp and that they satisfy the double chain condition.

We present a variant of the Diffie-Hellman scheme in which the number of bits exchanged is one third of what is used in the classical Diffie-Hellman scheme, while the offered security against attacks known today is the same. We also give applications for this variant and conjecture a extension of this variant further reducing the size of sent information.

We give necessary and sufficient conditions for two geometric Goppa codes C L (D, G) and C L (D, H) to be the same. As an application we characterize self-dual geometric Goppa codes.

This paper investigates the generalized rank weights, with a definition implied by the study of the generalized rank weight enumerator. We study rank metric codes over L, where L is a finite extension of a field K. This is a generalization of the case where K = F q and L = F q m of Gabidulin codes to arbitrary characteristic. We show equivalence to previous… (More)

An infinite series of curves is constructed in order to show that all linear codes can be obtained from curves using Goppa's construction. If one imposes conditions on the degree of the divisor used, then we derive criteria for linear codes to be algebraic-geometric. In particular, we investigate the family of q-ary Hamming codes, and prove that only those… (More)

Decoding geometric Goppa codes can be reduced to solving the key congruence of a received word in an affine ring. If the code length is smaller than the number of rational points on the curve, then this method can correct up to 1 2 (d * − 1) − s errors, where d * is the designed minimum distance of the code and s is the Clifford defect. The affine ring with… (More)