811 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
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
Weighing matrices and spherical codes
Mutually unbiased weighing matrices (MUWM) are closely related to an
antipodal spherical code with 4 angles. In the present paper, we clarify the
relationship between MUWM and the spherical sets, and give the complete
solution about the maximum size of a set of MUWM of weight 4 for any order.
Moreover we describe some natural generalization of a set of MUWM from the
viewpoint of spherical codes, and determine several maximum sizes of the
generalized sets. They include an affirmative answer of the problem of Best,
Kharaghani, and Ramp.Comment: Title is changed from "Association schemes related to weighing
matrices
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
Rigidity of spherical codes
A packing of spherical caps on the surface of a sphere (that is, a spherical
code) is called rigid or jammed if it is isolated within the space of packings.
In other words, aside from applying a global isometry, the packing cannot be
deformed. In this paper, we systematically study the rigidity of spherical
codes, particularly kissing configurations. One surprise is that the kissing
configuration of the Coxeter-Todd lattice is not jammed, despite being locally
jammed (each individual cap is held in place if its neighbors are fixed); in
this respect, the Coxeter-Todd lattice is analogous to the face-centered cubic
lattice in three dimensions. By contrast, we find that many other packings have
jammed kissing configurations, including the Barnes-Wall lattice and all of the
best kissing configurations known in four through twelve dimensions. Jamming
seems to become much less common for large kissing configurations in higher
dimensions, and in particular it fails for the best kissing configurations
known in 25 through 31 dimensions. Motivated by this phenomenon, we find new
kissing configurations in these dimensions, which improve on the records set in
1982 by the laminated lattices.Comment: 39 pages, 8 figure
On a generalization of distance sets
A subset in the -dimensional Euclidean space is called a -distance
set if there are exactly distinct distances between two distinct points in
and a subset is called a locally -distance set if for any point
in , there are at most distinct distances between and other points
in .
Delsarte, Goethals, and Seidel gave the Fisher type upper bound for the
cardinalities of -distance sets on a sphere in 1977. In the same way, we are
able to give the same bound for locally -distance sets on a sphere. In the
first part of this paper, we prove that if is a locally -distance set
attaining the Fisher type upper bound, then determining a weight function ,
is a tight weighted spherical -design. This result implies that
locally -distance sets attaining the Fisher type upper bound are
-distance sets. In the second part, we give a new absolute bound for the
cardinalities of -distance sets on a sphere. This upper bound is useful for
-distance sets for which the linear programming bound is not applicable. In
the third part, we discuss about locally two-distance sets in Euclidean spaces.
We give an upper bound for the cardinalities of locally two-distance sets in
Euclidean spaces. Moreover, we prove that the existence of a spherical
two-distance set in -space which attains the Fisher type upper bound is
equivalent to the existence of a locally two-distance set but not a
two-distance set in -space with more than points. We also
classify optimal (largest possible) locally two-distance sets for dimensions
less than eight. In addition, we determine the maximum cardinalities of locally
two-distance sets on a sphere for dimensions less than forty.Comment: 17 pages, 1 figur
A note on binary completely regular codes with large minimum distance
We classify all binary error correcting completely regular codes of length
with minimum distance .Comment: 4 page
Cubature formulas, geometrical designs, reproducing kernels, and Markov operators
Cubature formulas and geometrical designs are described in terms of
reproducing kernels for Hilbert spaces of functions on the one hand, and Markov
operators associated to orthogonal group representations on the other hand. In
this way, several known results for spheres in Euclidean spaces, involving
cubature formulas for polynomial functions and spherical designs, are shown to
generalize to large classes of finite measure spaces and
appropriate spaces of functions inside . The last section
points out how spherical designs are related to a class of reflection groups
which are (in general dense) subgroups of orthogonal groups
- …