14,757 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
The Treewidth of MDS and Reed-Muller Codes
The constraint complexity of a graphical realization of a linear code is the
maximum dimension of the local constraint codes in the realization. The
treewidth of a linear code is the least constraint complexity of any of its
cycle-free graphical realizations. This notion provides a useful
parametrization of the maximum-likelihood decoding complexity for linear codes.
In this paper, we prove the surprising fact that for maximum distance separable
codes and Reed-Muller codes, treewidth equals trelliswidth, which, for a code,
is defined to be the least constraint complexity (or branch complexity) of any
of its trellis realizations. From this, we obtain exact expressions for the
treewidth of these codes, which constitute the only known explicit expressions
for the treewidth of algebraic codes.Comment: This constitutes a major upgrade of previous versions; submitted to
IEEE Transactions on Information Theor
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