2,836 research outputs found
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
On Distance-Regular Graphs with Smallest Eigenvalue at Least
A non-complete geometric distance-regular graph is the point graph of a
partial geometry in which the set of lines is a set of Delsarte cliques. In
this paper, we prove that for fixed integer , there are only finitely
many non-geometric distance-regular graphs with smallest eigenvalue at least
, diameter at least three and intersection number
Optimal Networks from Error Correcting Codes
To address growth challenges facing large Data Centers and supercomputing
clusters a new construction is presented for scalable, high throughput, low
latency networks. The resulting networks require 1.5-5 times fewer switches,
2-6 times fewer cables, have 1.2-2 times lower latency and correspondingly
lower congestion and packet losses than the best present or proposed networks
providing the same number of ports at the same total bisection. These advantage
ratios increase with network size. The key new ingredient is the exact
equivalence discovered between the problem of maximizing network bisection for
large classes of practically interesting Cayley graphs and the problem of
maximizing codeword distance for linear error correcting codes. Resulting
translation recipe converts existent optimal error correcting codes into
optimal throughput networks.Comment: 14 pages, accepted at ANCS 2013 conferenc
Coding Theory and Algebraic Combinatorics
This chapter introduces and elaborates on the fruitful interplay of coding
theory and algebraic combinatorics, with most of the focus on the interaction
of codes with combinatorial designs, finite geometries, simple groups, sphere
packings, kissing numbers, lattices, and association schemes. In particular,
special interest is devoted to the relationship between codes and combinatorial
designs. We describe and recapitulate important results in the development of
the state of the art. In addition, we give illustrative examples and
constructions, and highlight recent advances. Finally, we provide a collection
of significant open problems and challenges concerning future research.Comment: 33 pages; handbook chapter, to appear in: "Selected Topics in
Information and Coding Theory", ed. by I. Woungang et al., World Scientific,
Singapore, 201
Problems on q-Analogs in Coding Theory
The interest in -analogs of codes and designs has been increased in the
last few years as a consequence of their new application in error-correction
for random network coding. There are many interesting theoretical, algebraic,
and combinatorial coding problems concerning these q-analogs which remained
unsolved. The first goal of this paper is to make a short summary of the large
amount of research which was done in the area mainly in the last few years and
to provide most of the relevant references. The second goal of this paper is to
present one hundred open questions and problems for future research, whose
solution will advance the knowledge in this area. The third goal of this paper
is to present and start some directions in solving some of these problems.Comment: arXiv admin note: text overlap with arXiv:0805.3528 by other author
Indexability, concentration, and VC theory
Degrading performance of indexing schemes for exact similarity search in high
dimensions has long since been linked to histograms of distributions of
distances and other 1-Lipschitz functions getting concentrated. We discuss this
observation in the framework of the phenomenon of concentration of measure on
the structures of high dimension and the Vapnik-Chervonenkis theory of
statistical learning.Comment: 17 pages, final submission to J. Discrete Algorithms (an expanded,
improved and corrected version of the SISAP'2010 invited paper, this e-print,
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