The minimum distance of codes in an array coming from telescopic semigroups

Abstract

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 semigroups only.

Generalized Hamming weights of q-ary Reed-Muller codes

“…In [40,29] bounds are given for the GHW's of algebraic geometry codes involving the gonality sequence. Good bounds are obtained [16,23] for the minimum distance of the one point AG codes in terms of the Weierstrass semigroup of the point at infinity, or what amounts to the same, the semigroup of the weight function ρ. This is generalized with the theory of Gröbner bases for affine algebraic sets in [33].…”

Towards a better understanding of the semigroup tree

“…Symmetric semigroups and their applications to coding theory have been studied, among others, in [1,4,6,7]. An important property of symmetric semigroups is that if c and g are the genus and the conductor of a symmetric semigroup Λ then any integer i is a gap of Λ if and only if c − 1 − i is a non-gap.…”

On the existence of order functions

“…Suppose that R has a basis consisting of a well-behaving sequence together with a surjective morphism ϕ : R → F n q of F q -algebras. In the following section a sketch will be given how this gives rise to a sequence of codes C(l) and a bound on the minimum distance of these codes following the ideas of [5,6,7,11,12,13,19,20,22,23]. This was generalized to a bound on the generalized Hamming weights by [10].…”

“…If R has a weight function, then the values of the weight function form a semigroup, and the bounds on the minimum distance of the codes C(l) can be formulated in terms of parameters of this semigroup. See [13].…”