José Ignacio Farrán

Goppa-like bounds for the generalized Feng--Rao distances

We give some general bounds and formulas for the generalized Feng-Rao distances (or generalized order bounds) in an arbitrary numerical semigroup. The obtained results can be regarded as generalizations of well-known facts on the classical Feng-Rao distance (or first order bound), namely its connection with the Goppa distance. These results show that their asymptotical behaviour is essentially the same as in the case of the classical order bound. Explicit computations are given for the second Feng-Rao distance.

On the parameters of algebraic-geometry codes related to Arf semigroups

We compute the order (or Feng-Rao (1994)) bound on the minimum distance of one-point algebraic-geometry codes C/sub /spl Omega//(P, /spl rho//sub t/Q), when the Weierstrass semigroup at the point Q is an Arf 91949) semigroup. The results developed to that purpose also provide the dimension of the improved geometric Goppa codes related to these C/sub /spl Omega// (P, /spl rho//sub t/Q).

On the Weight Hierarchy of Codes Coming From Semigroups With Two Generators

The weight hierarchy of one-point algebraic geometry codes can be estimated by means of the generalized order bounds, which are described in terms of a certain Weierstrass semigroup. The asymptotical behavior of such bounds for r ≥ 2 differs from that of the classical Feng-Rao distance (r=1) by the so-called Feng-Rao numbers. This paper is addressed to compute the Feng-Rao numbers for numerical semigroups of embedding dimension two (with two generators), obtaining a closed simple formula for the general case by using numerical semigroup techniques. These involve the computation of the Apéry set with respect to an integer of the semigroups under consideration. The formula obtained is applied to lower bounding the generalized Hamming weights, improving the bound given by Kirfel and Pellikaan in terms of the classical Feng-Rao distance. We also compare our bound with a modification of the Griesmer bound, improving this one in many cases.

Symbolic Hamburger-Noether expressions of plane curves and applications to AG codes

In this paper, we consider some practical applications of the symbolic Hamburger-Noether expressions for plane curves, which are introduced as a symbolic version of the so-called Hamburger-Noether expansions. More precisely, we give and develop in symbolic terms algorithms to compute the resolution tree of a plane curve (and the adjunction divisor, in particular), rational parametrizations for the branches of such a curve, special adjoints with assigned conditions (connected with different problems, like the so-called Brill-Noether algorithm), and the Weierstrass semigroup at P together with functions for each value in this semigroup, provided P is a rational branch of a singular plane model for the curve. Some other computational problems related to algebraic curves over perfect fields can be treated symbolically by means of such expressions, but we deal just with those connected with the effective construction and decoding of algebraic geometry codes.

On the generalized Feng-Rao numbers of numerical semigroups generated by intervals

We give some general results concerning the computation of the generalized Feng-Rao numbers of numerical semigroups. In the case of a numerical semigroup generated by an interval, a formula for the

The Second Feng-Rao Number for Codes Coming From Inductive Semigroups

r^{th}

Evaluation codes at singular points of algebraic differential equations

Feng-Rao number is obtained.

Gröbner basis for norm-trace codes

The second Feng-Rao number of every inductive numerical semigroup is explicitly computed. This number determines the asymptotical behavior of the order bound for the second Hamming weight of one-point algebraic geometry codes. In particular, this result is applied for the codes defined by asymptotically good towers of function fields whose Weierstrass semigroups are inductive. In addition, some properties of inductive numerical semigroups are studied, the involved Apéry sets are computed in a recursive way, and some tests to check whether given numerical semigroups are inductive or not are provided.

The second Feng–Rao number for codes coming from telescopic semigroups

We use the special geometry of singular points of algebraic differential equations on the affine plane over finite fields to study the main features and parameters of error correcting codes giving by evaluating functions at sets of singular points. In particular, one gets new methods to construct codes with designed minimum distance.

Goppa-like bounds for the generalized Feng-Rao distances

Heegard, Little and Saints worked out a Grobner basis algorithm for Hermitian codes. Here we extend such a result for codes on norm-trace curves.