517 research outputs found
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
Lower bound for the quantum capacity of a discrete memoryless quantum channel
We generalize the random coding argument of stabilizer codes and derive a
lower bound on the quantum capacity of an arbitrary discrete memoryless quantum
channel. For the depolarizing channel, our lower bound coincides with that
obtained by Bennett et al. We also slightly improve the quantum
Gilbert-Varshamov bound for general stabilizer codes, and establish an analogue
of the quantum Gilbert-Varshamov bound for linear stabilizer codes. Our proof
is restricted to the binary quantum channels, but its extension of to l-adic
channels is straightforward.Comment: 16 pages, REVTeX4. To appear in J. Math. Phys. A critical error in
fidelity calculation was corrected by using Hamada's result
(quant-ph/0112103). In the third version, we simplified formula and
derivation of the lower bound by proving p(Gamma)+q(Gamma)=1. In the second
version, we added an analogue of the quantum Gilbert-Varshamov bound for
linear stabilizer code
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
Asymptotic improvement of the Gilbert-Varshamov bound for linear codes
The Gilbert-Varshamov bound states that the maximum size A_2(n,d) of a binary
code of length n and minimum distance d satisfies A_2(n,d) >= 2^n/V(n,d-1)
where V(n,d) stands for the volume of a Hamming ball of radius d. Recently
Jiang and Vardy showed that for binary non-linear codes this bound can be
improved to A_2(n,d) >= cn2^n/V(n,d-1) for c a constant and d/n <= 0.499. In
this paper we show that certain asymptotic families of linear binary [n,n/2]
random double circulant codes satisfy the same improved Gilbert-Varshamov
bound.Comment: Submitted to IEEE Transactions on Information Theor
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
An Upper Bound on the Minimum Distance of LDPC Codes over GF(q)
In [1] a syndrome counting based upper bound on the minimum distance of
regular binary LDPC codes is given. In this paper we extend the bound to the
case of irregular and generalized LDPC codes over GF(q). The comparison to the
lower bound for LDPC codes over GF(q) and to the upper bound for non-binary
codes is done. The new bound is shown to lie under the Gilbert-Varshamov bound
at high rates.Comment: 4 pages, submitted to ISIT 201
End-to-End Error-Correcting Codes on Networks with Worst-Case Symbol Errors
The problem of coding for networks experiencing worst-case symbol errors is
considered. We argue that this is a reasonable model for highly dynamic
wireless network transmissions. We demonstrate that in this setup prior network
error-correcting schemes can be arbitrarily far from achieving the optimal
network throughput. A new transform metric for errors under the considered
model is proposed. Using this metric, we replicate many of the classical
results from coding theory. Specifically, we prove new Hamming-type,
Plotkin-type, and Elias-Bassalygo-type upper bounds on the network capacity. A
commensurate lower bound is shown based on Gilbert-Varshamov-type codes for
error-correction. The GV codes used to attain the lower bound can be
non-coherent, that is, they do not require prior knowledge of the network
topology. We also propose a computationally-efficient concatenation scheme. The
rate achieved by our concatenated codes is characterized by a Zyablov-type
lower bound. We provide a generalized minimum-distance decoding algorithm which
decodes up to half the minimum distance of the concatenated codes. The
end-to-end nature of our design enables our codes to be overlaid on the
classical distributed random linear network codes [1]. Furthermore, the
potentially intensive computation at internal nodes for the link-by-link
error-correction is un-necessary based on our design.Comment: Submitted for publication. arXiv admin note: substantial text overlap
with arXiv:1108.239
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