17 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
Self-orthogonal codes over a non-unital ring and combinatorial matrices
There is a local ring of order without identity for the
multiplication, defined by generators and relations as
We study a special construction of self-orthogonal codes over based on
combinatorial matrices related to two-class association schemes, Strongly
Regular Graphs (SRG), and Doubly Regular Tournaments (DRT).
We construct quasi self-dual codes over and Type IV codes, that is,
quasi self-dual codes whose all codewords have even Hamming weight. All these
codes can be represented as formally self-dual additive codes over \F_4. The
classical invariant theory bound for the weight enumerators of this class of
codesimproves the known bound on the minimum distance of Type IV codes over
Comment: 18 page
On relative -designs in polynomial association schemes
Motivated by the similarities between the theory of spherical -designs and
that of -designs in -polynomial association schemes, we study two
versions of relative -designs, the counterparts of Euclidean -designs for
- and/or -polynomial association schemes. We develop the theory based on
the Terwilliger algebra, which is a noncommutative associative semisimple
-algebra associated with each vertex of an association scheme. We
compute explicitly the Fisher type lower bounds on the sizes of relative
-designs, assuming that certain irreducible modules behave nicely. The two
versions of relative -designs turn out to be equivalent in the case of the
Hamming schemes. From this point of view, we establish a new algebraic
characterization of the Hamming schemes.Comment: 17 page