Numerical Semigroups and Codes

Abstract

A numerical semigroup is a subset of N containing 0, closed under addition and with finite complement in N. An important example of numerical semigroup is given by the Weierstrass semigroup at one point of a curve. In the theory of algebraic geometry codes, Weierstrass semigroups are crucial for defining bounds on the minimum distance as well as for defining improvements on the dimension of codes. We present these applications and some theoretical problems related to classification, characterization and counting of numerical semigroups.Notice that a symmetric numerical semigroup can not be pseudo-symmetric. Next lemma as well as its proof is analogous to Lemma 10. Lemma 12.A numerical semigroup Λ with odd conductor c is pseudo-symmetric if and only if for any non-negative integer i different from (c − 1)/2, if i is a gap, then c − 1 − i is a non-gap. 9

Ideals of Numerical Semigroups and Error-Correcting Codes

“…The point P ∞ = (0 : 1 : 0) is the unique point of H q at infinity. It can be proved (see, for instance, [1]…”

“…It is fundamental in the computation of bounds for the minimum distance of algebraic-geometry codes based on a single point as well as in the optimization of the redundancy of those codes. Its properties and applications can be seen in [7][8][9][10][11][12][13][14] and in the survey [1]. As a curiosity, it was proved in [7,15] that the set of elements of a numerical semigroup is determined by its ν sequence.…”

“…Ref. [1] is a survey on results related to the minimum distance, the error-correction capability, and the redundancy of the codes C B from the perspective of Weierstrass semigroups. In that case we considered, though, the dual codes < (…”

“…In a previous survey chapter [1], numerical semigroups were presented together with some of the related classical problems, and their importance for algebraic-geometry codes was explained. In particular, numerical semigroups can be used to establish decoding conditions, are useful to define bounds for the minimum distance of codes, and to improve the code dimension.…”

“…In this contribution, which is a continuation of that chapter, we will present some results relating ideals of numerical semigroups and the set of non-redundant parity-checks, the code length, the generalized Hamming weights, and the isometry-dual sequences of algebraic-geometry codes. The reader not familiar with algebraic geometry may be interested in the introductory sections of [1].…”