58 research outputs found
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
Recursive Code Construction for Random Networks
A modification of Koetter-Kschischang codes for random networks is presented
(these codes were also studied by Wang et al. in the context of authentication
problems). The new codes have higher information rate, while maintaining the
same error-correcting capabilities. An efficient error-correcting algorithm is
proposed for these codes.Comment: Submitted to IEEE Transactions on Information Theor
Data Dissemination Problem in Wireless Networks
In this work, we formulate and study a data dissemination problem, which can
be viewed as a generalization of the index coding problem and of the data
exchange problem to networks with an arbitrary topology. We define -solvable
networks, in which data dissemination can be achieved in communications
rounds. We show that the optimum number of transmissions for any one-round
communications scheme is given by the minimum rank of a certain constrained
family of matrices. For a special case of this problem, called bipartite data
dissemination problem, we present lower and upper graph-theoretic bounds on the
optimum number of transmissions. For general -solvable networks, we derive
an upper bound on the minimum number of transmissions in any scheme with rounds. We experimentally compare the obtained upper bound to a simple lower
bound.Comment: Notation clarificatio
Refined Upper Bounds on Stopping Redundancy of Binary Linear Codes
The -th stopping redundancy of the binary
code , , is defined as the minimum number of rows in
the parity-check matrix of , such that the smallest stopping set is
of size at least . The stopping redundancy is defined as
. In this work, we improve on the probabilistic analysis of
stopping redundancy, proposed by Han, Siegel and Vardy, which yields the best
bounds known today. In our approach, we judiciously select the first few rows
in the parity-check matrix, and then continue with the probabilistic method. By
using similar techniques, we improve also on the best known bounds on
, for . Our approach is compared to the
existing methods by numerical computations.Comment: 5 pages; ITW 201
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
Error-Correction in Flash Memories via Codes in the Ulam Metric
We consider rank modulation codes for flash memories that allow for handling
arbitrary charge-drop errors. Unlike classical rank modulation codes used for
correcting errors that manifest themselves as swaps of two adjacently ranked
elements, the proposed \emph{translocation rank codes} account for more general
forms of errors that arise in storage systems. Translocations represent a
natural extension of the notion of adjacent transpositions and as such may be
analyzed using related concepts in combinatorics and rank modulation coding.
Our results include derivation of the asymptotic capacity of translocation rank
codes, construction techniques for asymptotically good codes, as well as simple
decoding methods for one class of constructed codes. As part of our exposition,
we also highlight the close connections between the new code family and
permutations with short common subsequences, deletion and insertion
error-correcting codes for permutations, and permutation codes in the Hamming
distance
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