6,236 research outputs found
An asymptotic existence result on compressed sensing matrices
For any rational number and all sufficiently large we give a
deterministic construction for an compressed
sensing matrix with -recoverability where . Our
method uses pairwise balanced designs and complex Hadamard matrices in the
construction of -equiangular frames, which we introduce as a
generalisation of equiangular tight frames. The method is general and produces
good compressed sensing matrices from any appropriately chosen pairwise
balanced design. The -recoverability performance is specified as a
simple function of the parameters of the design. To obtain our asymptotic
existence result we prove new results on the existence of pairwise balanced
designs in which the numbers of blocks of each size are specified.Comment: 15 pages, no figures. Minor improvements and updates in February 201
All or Nothing at All
We continue a study of unconditionally secure all-or-nothing transforms
(AONT) begun in \cite{St}. An AONT is a bijective mapping that constructs s
outputs from s inputs. We consider the security of t inputs, when s-t outputs
are known. Previous work concerned the case t=1; here we consider the problem
for general t, focussing on the case t=2. We investigate constructions of
binary matrices for which the desired properties hold with the maximum
probability. Upper bounds on these probabilities are obtained via a quadratic
programming approach, while lower bounds can be obtained from combinatorial
constructions based on symmetric BIBDs and cyclotomy. We also report some
results on exhaustive searches and random constructions for small values of s.Comment: 23 page
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
Coding-Theoretic Methods for Sparse Recovery
We review connections between coding-theoretic objects and sparse learning
problems. In particular, we show how seemingly different combinatorial objects
such as error-correcting codes, combinatorial designs, spherical codes,
compressed sensing matrices and group testing designs can be obtained from one
another. The reductions enable one to translate upper and lower bounds on the
parameters attainable by one object to another. We survey some of the
well-known reductions in a unified presentation, and bring some existing gaps
to attention. New reductions are also introduced; in particular, we bring up
the notion of minimum "L-wise distance" of codes and show that this notion
closely captures the combinatorial structure of RIP-2 matrices. Moreover, we
show how this weaker variation of the minimum distance is related to
combinatorial list-decoding properties of codes.Comment: Added Lemma 34 in the first revision. Original version in Proceedings
of the Allerton Conference on Communication, Control and Computing, September
201
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