30 research outputs found
New proofs of the Assmus-Mattson theorem based on the Terwilliger algebra
We use the Terwilliger algebra to provide a new approach to the
Assmus-Mattson theorem. This approach also includes another proof of the
minimum distance bound shown by Martin as well as its dual.Comment: 15 page
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
Constructions of new matroids and designs over GF(q)
A perfect matroid design (PMD) is a matroid whose flats of the same rank all
have the same size. In this paper we introduce the q-analogue of a PMD and its
properties. In order to do that, we first establish a new cryptomorphic
definition for q-matroids. We show that q-Steiner systems are examples of
q-PMD's and we use this q-matroid structure to construct subspace designs from
q-Steiner systems. We apply this construction to S(2, 3, 13; q) q-Steiner
systems and hence establish the existence of subspace designs with previously
unknown parameters
The Projectivization Matroid of a -Matroid
In this paper, we investigate the relation between a -matroid and its
associated matroid called the projectivization matroid. The latter arises by
projectivizing the groundspace of the -matroid, and considering the
projective space as the groundset of the associated matroid, on which is
defined a rank function compatible with that of the -matroid. We show that
the projectivization map is a functor from categories of -matroids to
categories of matroids. This relation is used to prove new results about maps
of -matroids. Furthermore, we show the characteristic polynomial of a
-matroid is equal to that of the projectivization matroid, which we use to
establish a recursive formula for the characteristic polynomial of a
-matroid in terms of the characteristic polynomial of its minors. Finally we
use the projectivization matroid to prove a -analogue of the critical
theorem in terms of -linear rank metric codes and
-matroids