6 research outputs found
Commutative association schemes
Association schemes were originally introduced by Bose and his co-workers in
the design of statistical experiments. Since that point of inception, the
concept has proved useful in the study of group actions, in algebraic graph
theory, in algebraic coding theory, and in areas as far afield as knot theory
and numerical integration. This branch of the theory, viewed in this collection
of surveys as the "commutative case," has seen significant activity in the last
few decades. The goal of the present survey is to discuss the most important
new developments in several directions, including Gelfand pairs, cometric
association schemes, Delsarte Theory, spin models and the semidefinite
programming technique. The narrative follows a thread through this list of
topics, this being the contrast between combinatorial symmetry and
group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes
(based on group actions) and its connection to the Terwilliger algebra (based
on combinatorial symmetry). We propose this new role of the Terwilliger algebra
in Delsarte Theory as a central topic for future work.Comment: 36 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
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