334 research outputs found
Improved Nearly-MDS Expander Codes
A construction of expander codes is presented with the following three
properties:
(i) the codes lie close to the Singleton bound, (ii) they can be encoded in
time complexity that is linear in their code length, and (iii) they have a
linear-time bounded-distance decoder.
By using a version of the decoder that corrects also erasures, the codes can
replace MDS outer codes in concatenated constructions, thus resulting in
linear-time encodable and decodable codes that approach the Zyablov bound or
the capacity of memoryless channels. The presented construction improves on an
earlier result by Guruswami and Indyk in that any rate and relative minimum
distance that lies below the Singleton bound is attainable for a significantly
smaller alphabet size.Comment: Part of this work was presented at the 2004 IEEE Int'l Symposium on
Information Theory (ISIT'2004), Chicago, Illinois (June 2004). This work was
submitted to IEEE Transactions on Information Theory on January 21, 2005. To
appear in IEEE Transactions on Information Theory, August 2006. 12 page
Correcting a Fraction of Errors in Nonbinary Expander Codes with Linear Programming
A linear-programming decoder for \emph{nonbinary} expander codes is
presented. It is shown that the proposed decoder has the maximum-likelihood
certificate properties. It is also shown that this decoder corrects any pattern
of errors of a relative weight up to approximately 1/4 \delta_A \delta_B (where
\delta_A and \delta_B are the relative minimum distances of the constituent
codes).Comment: Part of this work was presented at the IEEE International Symposium
on Information Theory 2009, Seoul, Kore
Mathematical Programming Decoding of Binary Linear Codes: Theory and Algorithms
Mathematical programming is a branch of applied mathematics and has recently
been used to derive new decoding approaches, challenging established but often
heuristic algorithms based on iterative message passing. Concepts from
mathematical programming used in the context of decoding include linear,
integer, and nonlinear programming, network flows, notions of duality as well
as matroid and polyhedral theory. This survey article reviews and categorizes
decoding methods based on mathematical programming approaches for binary linear
codes over binary-input memoryless symmetric channels.Comment: 17 pages, submitted to the IEEE Transactions on Information Theory.
Published July 201
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