35,885 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
Quickest Sequence Phase Detection
A phase detection sequence is a length- cyclic sequence, such that the
location of any length- contiguous subsequence can be determined from a
noisy observation of that subsequence. In this paper, we derive bounds on the
minimal possible in the limit of , and describe some sequence
constructions. We further consider multiple phase detection sequences, where
the location of any length- contiguous subsequence of each sequence can be
determined simultaneously from a noisy mixture of those subsequences. We study
the optimal trade-offs between the lengths of the sequences, and describe some
sequence constructions. We compare these phase detection problems to their
natural channel coding counterparts, and show a strict separation between the
fundamental limits in the multiple sequence case. Both adversarial and
probabilistic noise models are addressed.Comment: To appear in the IEEE Transactions on Information Theor
Critical Noise Levels for LDPC decoding
We determine the critical noise level for decoding low density parity check
error correcting codes based on the magnetization enumerator (\cM), rather
than on the weight enumerator (\cW) employed in the information theory
literature. The interpretation of our method is appealingly simple, and the
relation between the different decoding schemes such as typical pairs decoding,
MAP, and finite temperature decoding (MPM) becomes clear. In addition, our
analysis provides an explanation for the difference in performance between MN
and Gallager codes. Our results are more optimistic than those derived via the
methods of information theory and are in excellent agreement with recent
results from another statistical physics approach.Comment: 9 pages, 5 figure
Recovery thresholds in the sparse planted matching problem
We consider the statistical inference problem of recovering an unknown
perfect matching, hidden in a weighted random graph, by exploiting the
information arising from the use of two different distributions for the weights
on the edges inside and outside the planted matching. A recent work has
demonstrated the existence of a phase transition, in the large size limit,
between a full and a partial recovery phase for a specific form of the weights
distribution on fully connected graphs. We generalize and extend this result in
two directions: we obtain a criterion for the location of the phase transition
for generic weights distributions and possibly sparse graphs, exploiting a
technical connection with branching random walk processes, as well as a
quantitatively more precise description of the critical regime around the phase
transition.Comment: 19 pages, 8 figure
Voronoi regions for binary linear block codes
The Voronoi regions of a block code govern many aspects of the code's performance on a Gaussian channel, and they are fundamental instruments in, for example, error probability analysis and soft-decision decoding. The article presents an efficient method for finding the boundaries of the Voronoi regions for an arbitrary binary linear block code. Two theoretical results together lead to the Voronoi regions. First, it is shown that the question of the Voronoi neighborship can be reduced into testing a simpler relation, called the Gabriel neighborship. Second, a fast method of recognising Gabriel neighbors is proposed. These results are finally employed to describe the Voronoi regions for the Golay codes and several BCH codes, including Hamming codes
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