88 research outputs found
Directed Graph Representation of Half-Rate Additive Codes over GF(4)
We show that (n,2^n) additive codes over GF(4) can be represented as directed
graphs. This generalizes earlier results on self-dual additive codes over
GF(4), which correspond to undirected graphs. Graph representation reduces the
complexity of code classification, and enables us to classify additive (n,2^n)
codes over GF(4) of length up to 7. From this we also derive classifications of
isodual and formally self-dual codes. We introduce new constructions of
circulant and bordered circulant directed graph codes, and show that these
codes will always be isodual. A computer search of all such codes of length up
to 26 reveals that these constructions produce many codes of high minimum
distance. In particular, we find new near-extremal formally self-dual codes of
length 11 and 13, and isodual codes of length 24, 25, and 26 with better
minimum distance than the best known self-dual codes.Comment: Presented at International Workshop on Coding and Cryptography (WCC
2009), 10-15 May 2009, Ullensvang, Norway. (14 pages, 2 figures
Self-dual codes, subcode structures, and applications.
The classification of self-dual codes has been an extremely active area in coding theory since 1972 [33]. A particularly interesting class of self-dual codes is those of Type II which have high minimum distance (called extremal or near-extremal). It is notable that this class of codes contains famous unique codes: the extended Hamming [8,4,4] code, the extended Golay [24,12,8] code, and the extended quadratic residue [48,24,12] code. We examine the subcode structures of Type II codes for lengths up to 24, extremal Type II codes of length 32, and give partial results on the extended quadratic residue [48,24,12] code. We also develop a generalization of self-dual codes to Network Coding Theory and give some results on existence of self-dual network codes with largest minimum distance for lengths up to 10. Complementary Information Set (CIS for short) codes, a class of classical codes recently developed in [7], have important applications to Cryptography. CIS codes contain self-dual codes as a subclass. We give a new classification result for CIS codes of length 14 and a partial result for length 16
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
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
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