191 research outputs found

    Generalized Random Gilbert-Varshamov Codes: Typical Error Exponent and Concentration Properties

    Get PDF
    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

    Full text link
    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

    Get PDF
    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

    Full text link
    Given positive integers nn and dd, let A2(n,d)A_2(n,d) denote the maximum size of a binary code of length nn and minimum distance dd. The well-known Gilbert-Varshamov bound asserts that A2(n,d)≄2n/V(n,d−1)A_2(n,d) \geq 2^n/V(n,d-1), where V(n,d)=∑i=0d(ni)V(n,d) = \sum_{i=0}^{d} {n \choose i} is the volume of a Hamming sphere of radius dd. We show that, in fact, there exists a positive constant cc such that A2(n,d)≄c2nV(n,d−1)log⁥2V(n,d−1) A_2(n,d) \geq c \frac{2^n}{V(n,d-1)} \log_2 V(n,d-1) whenever d/n≀0.499d/n \le 0.499. 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

    Get PDF
    © 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

    Full text link
    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

    Full text link
    We investigate the asymptotic rates of length-nn binary codes with VC-dimension at most dndn and minimum distance at least ÎŽn\delta n. 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
    • 

    corecore