2,087 research outputs found
Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs
A novel technique, based on the pseudo-random properties of certain graphs known as expanders, is used to obtain novel simple explicit constructions of asymptotically good codes. In one of the constructions, the expanders are used to enhance Justesen codes by replicating, shuffling, and then regrouping the code coordinates. For any fixed (small) rate, and for a sufficiently large alphabet, the codes thus obtained lie above the Zyablov bound. Using these codes as outer codes in a concatenated scheme, a second asymptotic good construction is obtained which applies to small alphabets (say, GF(2)) as well. Although these concatenated codes lie below the Zyablov bound, they are still superior to previously known explicit constructions in the zero-rate neighborhood
Deletion codes in the high-noise and high-rate regimes
The noise model of deletions poses significant challenges in coding theory,
with basic questions like the capacity of the binary deletion channel still
being open. In this paper, we study the harder model of worst-case deletions,
with a focus on constructing efficiently decodable codes for the two extreme
regimes of high-noise and high-rate. Specifically, we construct polynomial-time
decodable codes with the following trade-offs (for any eps > 0):
(1) Codes that can correct a fraction 1-eps of deletions with rate poly(eps)
over an alphabet of size poly(1/eps);
(2) Binary codes of rate 1-O~(sqrt(eps)) that can correct a fraction eps of
deletions; and
(3) Binary codes that can be list decoded from a fraction (1/2-eps) of
deletions with rate poly(eps)
Our work is the first to achieve the qualitative goals of correcting a
deletion fraction approaching 1 over bounded alphabets, and correcting a
constant fraction of bit deletions with rate aproaching 1. The above results
bring our understanding of deletion code constructions in these regimes to a
similar level as worst-case errors
Linear-Codes-Based Lossless Joint Source-Channel Coding for Multiple-Access Channels
A general lossless joint source-channel coding (JSCC) scheme based on linear
codes and random interleavers for multiple-access channels (MACs) is presented
and then analyzed in this paper. By the information-spectrum approach and the
code-spectrum approach, it is shown that a linear code with a good joint
spectrum can be used to establish limit-approaching lossless JSCC schemes for
correlated general sources and general MACs, where the joint spectrum is a
generalization of the input-output weight distribution. Some properties of
linear codes with good joint spectra are investigated. A formula on the
"distance" property of linear codes with good joint spectra is derived, based
on which, it is further proved that, the rate of any systematic codes with good
joint spectra cannot be larger than the reciprocal of the corresponding
alphabet cardinality, and any sparse generator matrices cannot yield linear
codes with good joint spectra. The problem of designing arbitrary rate coding
schemes is also discussed. A novel idea called "generalized puncturing" is
proposed, which makes it possible that one good low-rate linear code is enough
for the design of coding schemes with multiple rates. Finally, various coding
problems of MACs are reviewed in a unified framework established by the
code-spectrum approach, under which, criteria and candidates of good linear
codes in terms of spectrum requirements for such problems are clearly
presented.Comment: 18 pages, 3 figure
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