Carlos Munuera

Fast communication: Steganography and error-correcting codes

We show some relations between steganographic algorithms and error-correcting codes. By using these relations we give a method to construct good steganographic protocols and deduce their properties from those of the corresponding codes.

On the generalized Hamming weights of geometric Goppa codes

We study the generalized Hamming weights of geometric Goppa codes, giving a translation of these weights into arithmetical terms, concerning the arithmetic of the curve used to construct the code. We derive some results and bounds that we apply to some known codes. >

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.

The Weight Hierarchy of Hermitian Codes

We compute the complete weight hierarchies of all Hermitian codes. The tools used are the arithmetic of Hermitian curves and the order bound on the generalized Hamming weights.

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 order bounds for one-point AG codes

The order bound for the minimum distance of algebraic geometry codes was originally defined for the duals of one-point codes and later generalized for arbitrary algebraic geometry codes. Another bound of order type for the minimum distance of general linear codes, and for codes from order domains in particular, was given in [H. Andersen and O. Geil, Evaluation codes from order domain theory, Finite Fields and their Applications 14 (2008), pp. 92-123]. Here we investigate in detail the application of that bound to one-point algebraic geometry codes, obtaining a bound

The second and third generalized Hamming weights of Hermitian codes

d^*

Equality of geometric Goppa codes and equivalence of divisors

for the minimum distance of these codes. We establish a connection between

Castle curves and codes

d^*

On the main conjecture on geometric MDS codes

and the order bound and its generalizations. We also study the improved code constructions based on