60 research outputs found
Linear complexity universal decoding with exponential error probability decay
In this manuscript we consider linear complexity binary linear block encoders and decoders that operate universally with exponential error probability decay. Such scenarios may be relevant in wireless scenarios where probability distributions may not be fully characterized due to the dynamic nature of wireless environments. More specifically, we consider the setting of fixed length-to-fixed length near-lossless data compression of a memoryless binary source of unknown probability distribution as well as the dual setting of communicating on a binary symmetric channel (BSC) with unknown crossover probability. We introduce a new 'min-max distance' metric, analogous to minimum distance, that addresses the universal binary setting and has the same properties as that of minimum distance on BSCs with known crossover probability. The code construction and decoding algorithm are universal extensions of the 'expander codes' framework of Barg and Zemor and have identical complexity and exponential error probability performance
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
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
Probabilistic Model Counting with Short XORs
The idea of counting the number of satisfying truth assignments (models) of a
formula by adding random parity constraints can be traced back to the seminal
work of Valiant and Vazirani, showing that NP is as easy as detecting unique
solutions. While theoretically sound, the random parity constraints in that
construction have the following drawback: each constraint, on average, involves
half of all variables. As a result, the branching factor associated with
searching for models that also satisfy the parity constraints quickly gets out
of hand. In this work we prove that one can work with much shorter parity
constraints and still get rigorous mathematical guarantees, especially when the
number of models is large so that many constraints need to be added. Our work
is based on the realization that the essential feature for random systems of
parity constraints to be useful in probabilistic model counting is that the
geometry of their set of solutions resembles an error-correcting code.Comment: To appear in SAT 1
Improved Decoding of Expander Codes
We study the classical expander codes, introduced by Sipser and Spielman [M. Sipser and D. A. Spielman, 1996]. Given any constants 0 < ?, ? < 1/2, and an arbitrary bipartite graph with N vertices on the left, M < N vertices on the right, and left degree D such that any left subset S of size at most ? N has at least (1-?)|S|D neighbors, we show that the corresponding linear code given by parity checks on the right has distance at least roughly {? N}/{2 ?}. This is strictly better than the best known previous result of 2(1-?) ? N [Madhu Sudan, 2000; Viderman, 2013] whenever ? < 1/2, and improves the previous result significantly when ? is small. Furthermore, we show that this distance is tight in general, thus providing a complete characterization of the distance of general expander codes.
Next, we provide several efficient decoding algorithms, which vastly improve previous results in terms of the fraction of errors corrected, whenever ? < 1/4. Finally, we also give a bound on the list-decoding radius of general expander codes, which beats the classical Johnson bound in certain situations (e.g., when the graph is almost regular and the code has a high rate).
Our techniques exploit novel combinatorial properties of bipartite expander graphs. In particular, we establish a new size-expansion tradeoff, which may be of independent interests
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