8 research outputs found
Feng-Rao decoding of primary codes
We show that the Feng-Rao bound for dual codes and a similar bound by
Andersen and Geil [H.E. Andersen and O. Geil, Evaluation codes from order
domain theory, Finite Fields Appl., 14 (2008), pp. 92-123] for primary codes
are consequences of each other. This implies that the Feng-Rao decoding
algorithm can be applied to decode primary codes up to half their designed
minimum distance. The technique applies to any linear code for which
information on well-behaving pairs is available. Consequently we are able to
decode efficiently a large class of codes for which no non-trivial decoding
algorithm was previously known. Among those are important families of
multivariate polynomial codes. Matsumoto and Miura in [R. Matsumoto and S.
Miura, On the Feng-Rao bound for the L-construction of algebraic geometry
codes, IEICE Trans. Fundamentals, E83-A (2000), pp. 926-930] (See also [P.
Beelen and T. H{\o}holdt, The decoding of algebraic geometry codes, in Advances
in algebraic geometry codes, pp. 49-98]) derived from the Feng-Rao bound a
bound for primary one-point algebraic geometric codes and showed how to decode
up to what is guaranteed by their bound. The exposition by Matsumoto and Miura
requires the use of differentials which was not needed in [Andersen and Geil
2008]. Nevertheless we demonstrate a very strong connection between Matsumoto
and Miura's bound and Andersen and Geil's bound when applied to primary
one-point algebraic geometric codes.Comment: elsarticle.cls, 23 pages, no figure. Version 3 added citations to the
works by I.M. Duursma and R. Pellikaa
Distance bounds for algebraic geometric codes
Various methods have been used to obtain improvements of the Goppa lower
bound for the minimum distance of an algebraic geometric code. The main methods
divide into two categories and all but a few of the known bounds are special
cases of either the Lundell-McCullough floor bound or the Beelen order bound.
The exceptions are recent improvements of the floor bound by
Guneri-Stichtenoth-Taskin, and Duursma-Park, and of the order bound by
Duursma-Park and Duursma-Kirov. In this paper we provide short proofs for all
floor bounds and most order bounds in the setting of the van Lint and Wilson AB
method. Moreover, we formulate unifying theorems for order bounds and formulate
the DP and DK order bounds as natural but different generalizations of the
Feng-Rao bound for one-point codes.Comment: 29 page
Two-point AG codes from one of the Skabelund maximal curves
In this paper, we investigate two-point Algebraic Geometry codes associated
to the Skabelund maximal curve constructed as a cyclic cover of the Suzuki
curve. In order to estimate the minimum distance of such codes, we make use of
the generalized order bound introduced by P. Beelen and determine certain
two-point Weierstrass semigroups of the curve.Comment: 15 page
An Introduction to Algebraic Geometry codes
We present an introduction to the theory of algebraic geometry codes.
Starting from evaluation codes and codes from order and weight functions,
special attention is given to one-point codes and, in particular, to the family
of Castle codes
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 [1]. Here we investigate in detail the application of that bound to one-point algebraic geometry codes, obtaining a bound d* for the minimum distance of these codes. We establish a connection between d* and the order bound and its generalizations. We also study the improved code constructions based on d*. Finally we extend d* to all generalized Hamming weights.53489504Danish National Science Research Council [FNV-21040368]Danish FNU [272-07-0266]Junta de CyL [VA065A07]Spanish Ministry for Science and Technology [MTM-2007-66842-C02-01, MTM 2007-64704]Aalborg UniversityThe Technical University of DenmarkDanish National Science Research Council [FNV-21040368]Danish FNU [272-07-0266]Junta de CyL [VA065A07]Spanish Ministry for Science and Technology [MTM-2007-66842-C02-01, MTM 2007-64704
Pesos de Hamming de c贸digos Castillo
C贸digos Castillo son c贸digos algebraico geom茅tricos unipuntuales sobre curvas Castillo. Esta Familia contiene algunos de los c贸digos AG m谩s importantes entre los estudiados en la literatura hasta la fecha. En esta tesis se obtiene una caracterizaci贸n expl铆cita sobre las estimaciones de la distancia m铆nima y los pesos de Hamming generalizados de los c贸digos Castillo.Departamento de Algebra, An谩lisis Matem谩tico, Geometr铆a y Topolog铆