191 research outputs found
Generalized Random Gilbert-Varshamov Codes: Typical Error Exponent and Concentration Properties
We find the exact typical error exponent of constant composition generalized
random Gilbert-Varshamov (RGV) codes over DMCs channels with generalized
likelihood decoding. We show that the typical error exponent of the RGV
ensemble is equal to the expurgated error exponent, provided that the RGV
codebook parameters are chosen appropriately. We also prove that the random
coding exponent converges in probability to the typical error exponent, and the
corresponding non-asymptotic concentration rates are derived. Our results show
that the decay rate of the lower tail is exponential while that of the upper
tail is double exponential above the expurgated error exponent. The explicit
dependence of the decay rates on the RGV distance functions is characterized.Comment: 60 pages, 2 figure
Subquadratic time encodable codes beating the Gilbert-Varshamov bound
We construct explicit algebraic geometry codes built from the
Garcia-Stichtenoth function field tower beating the Gilbert-Varshamov bound for
alphabet sizes at least 192. Messages are identied with functions in certain
Riemann-Roch spaces associated with divisors supported on multiple places.
Encoding amounts to evaluating these functions at degree one places. By
exploiting algebraic structures particular to the Garcia-Stichtenoth tower, we
devise an intricate deterministic \omega/2 < 1.19 runtime exponent encoding and
1+\omega/2 < 2.19 expected runtime exponent randomized (unique and list)
decoding algorithms. Here \omega < 2.373 is the matrix multiplication exponent.
If \omega = 2, as widely believed, the encoding and decoding runtimes are
respectively nearly linear and nearly quadratic. Prior to this work, encoding
(resp. decoding) time of code families beating the Gilbert-Varshamov bound were
quadratic (resp. cubic) or worse
The problem with the SURF scheme
There is a serious problem with one of the assumptions made in the security
proof of the SURF scheme. This problem turns out to be easy in the regime of
parameters needed for the SURF scheme to work.
We give afterwards the old version of the paper for the reader's convenience.Comment: Warning : we found a serious problem in the security proof of the
SURF scheme. We explain this problem here and give the old version of the
paper afterward
Asymptotic Improvement of the Gilbert-Varshamov Bound on the Size of Binary Codes
Given positive integers and , let denote the maximum size
of a binary code of length and minimum distance . The well-known
Gilbert-Varshamov bound asserts that , where
is the volume of a Hamming sphere of
radius . We show that, in fact, there exists a positive constant such
that whenever . The result follows by recasting the Gilbert- Varshamov bound into a
graph-theoretic framework and using the fact that the corresponding graph is
locally sparse. Generalizations and extensions of this result are briefly
discussed.Comment: 10 pages, 3 figures; to appear in the IEEE Transactions on
Information Theory, submitted August 12, 2003, revised March 28, 200
Generalized Random Gilbert-Varshamov Codes
© 1963-2012 IEEE. We introduce a random coding technique for transmission over discrete memoryless channels, reminiscent of the basic construction attaining the Gilbert-Varshamov bound for codes in Hamming spaces. The code construction is based on drawing codewords recursively from a fixed type class, in such a way that a newly generated codeword must be at a certain minimum distance from all previously chosen codewords, according to some generic distance function. We derive an achievable error exponent for this construction and prove its tightness with respect to the ensemble average. We show that the exponent recovers the Csiszår and Körner exponent as a special case, which is known to be at least as high as both the random-coding and expurgated exponents, and we establish the optimality of certain choices of the distance function. In addition, for additive distances and decoding metrics, we present an equivalent dual expression, along with a generalization to infinite alphabets via cost-constrained random coding.ER
Minimum Distance Distribution of Irregular Generalized LDPC Code Ensembles
In this paper, the minimum distance distribution of irregular generalized
LDPC (GLDPC) code ensembles is investigated. Two classes of GLDPC code
ensembles are analyzed; in one case, the Tanner graph is regular from the
variable node perspective, and in the other case the Tanner graph is completely
unstructured and irregular. In particular, for the former ensemble class we
determine exactly which ensembles have minimum distance growing linearly with
the block length with probability approaching unity with increasing block
length. This work extends previous results concerning LDPC and regular GLDPC
codes to the case where a hybrid mixture of check node types is used.Comment: 5 pages, 1 figure. Submitted to the IEEE International Symposium on
Information Theory (ISIT) 201
On the VC-Dimension of Binary Codes
We investigate the asymptotic rates of length- binary codes with
VC-dimension at most and minimum distance at least . Two upper
bounds are obtained, one as a simple corollary of a result by Haussler and the
other via a shortening approach combining Sauer-Shelah lemma and the linear
programming bound. Two lower bounds are given using Gilbert-Varshamov type
arguments over constant-weight and Markov-type sets
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