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Permutation decoding and the stopping redundancy hierarchy of linear block codes
 in Proc. IEEE International Symp. Information Theory
, 2007
"... We introduce the notion of the stopping redundancy hierarchy of a linear block code as a measure of the tradeoff between performance and complexity of iterative decoding for the binary erasure channel. We derive lower and upper bounds for the stopping redundancy hierarchy via Lovász’s Local Lemma a ..."
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We introduce the notion of the stopping redundancy hierarchy of a linear block code as a measure of the tradeoff between performance and complexity of iterative decoding for the binary erasure channel. We derive lower and upper bounds for the stopping redundancy hierarchy via Lovász’s Local Lemma and Bonferronitype inequalities, and specialize them for codes with cyclic paritycheck matrices. Based on the observed properties of paritycheck matrices with good stopping redundancy characteristics, we develop a novel decoding technique, termed automorphism group decoding, that combines iterative message passing and permutation decoding. We also present bounds on the smallest number of permutations of an automorphism group decoder needed to correct any set of erasures up to a prescribed size. Simulation results demonstrate that for a large number of algebraic codes, the performance of the new decoding method is close to that of maximum likelihood decoding.
Bounds on singleexclusion numbers and stopping redundancy of MDS codes
 IN PROC. IEEE INTERNATIONAL SYMP. INFORMATION THEORY
, 2007
"... New bounds on singleexclusion numbers are obtained via probabilistic arguments, recurrent relations, as well as explicit constructions. The new bounds are used to better understand the stopping redundancy of MDS codes. In particular, it is shown that for any fixed k, the stopping ´ redundancy ` ´ ..."
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New bounds on singleexclusion numbers are obtained via probabilistic arguments, recurrent relations, as well as explicit constructions. The new bounds are used to better understand the stopping redundancy of MDS codes. In particular, it is shown that for any fixed k, the stopping ´ redundancy ` ´ of ´ a 1 and 1 + o(1). linear [n, k] MDS code is between 1 `n k+1 k I.
An Upper Bound on the Separating Redundancy of Linear Block Codes
"... Abstract—Linear block codes over noisy channels causing both erasures and errors can be decoded by deleting the erased symbols and decoding the resulting vector with respect to a punctured code and then retrieving the erased symbols. This can be accomplished using separating paritycheck matrices. F ..."
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Abstract—Linear block codes over noisy channels causing both erasures and errors can be decoded by deleting the erased symbols and decoding the resulting vector with respect to a punctured code and then retrieving the erased symbols. This can be accomplished using separating paritycheck matrices. For a given maximum number of correctable erasures, such matrices yield paritycheck equations that do not check any of the erased symbols and which are sufficient to characterize all punctured codes corresponding to this maximum number of erasures. Separating paritycheck matrices typically have redundant rows. An upper bound on the minimum number of rows in separating paritycheck matrices, which is called the separating redundancy, is derived which proves that the separating redundancy tends to behave linearly as a function of the code length. I.
Permutation Decoding and the Stopping Redundancy Hierarchy of Cyclic and Extended Cyclic Codes
"... Abstract — We introduce the notion of the stopping redundancy hierarchy of a linear block code as a measure of the tradeoff between performance and complexity of iterative decoding for the binary erasure channel. We derive lower and upper bounds for the stopping redundancy hierarchy via Lovász’s L ..."
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Abstract — We introduce the notion of the stopping redundancy hierarchy of a linear block code as a measure of the tradeoff between performance and complexity of iterative decoding for the binary erasure channel. We derive lower and upper bounds for the stopping redundancy hierarchy via Lovász’s Local Lemma and Bonferronitype inequalities, and specialize them for codes with cyclic paritycheck matrices. Based on the observed properties of paritycheck matrices with good stopping redundancy characteristics, we develop a novel decoding technique, termed automorphism group decoding, that combines iterative message passing and permutation decoding. We also present bounds on the smallest number of permutations of an automorphism group decoder needed to correct any set of erasures up to a prescribed size. Simulation results demonstrate that for a large number of algebraic codes, the performance of the new decoding method is close to that of maximum likelihood decoding.
Stopping Sets of Algebraic Geometry Codes
"... Abstract — Stopping sets and stopping set distribution of a linear code play an important role in the performance analysis of iterative decoding for this linear code. Let C be an [n, k] linear code over Fq with paritycheck matrix H, wheretherowsof H may be dependent. Let [n] ={1, 2,...,n} denote th ..."
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Abstract — Stopping sets and stopping set distribution of a linear code play an important role in the performance analysis of iterative decoding for this linear code. Let C be an [n, k] linear code over Fq with paritycheck matrix H, wheretherowsof H may be dependent. Let [n] ={1, 2,...,n} denote the set of column indices of H. Astopping set S of C with paritycheck matrix H is a subset of [n] such that the restriction of H to S does not contain a row of weight 1. The stopping set distribution {Ti (H)} n i=0 enumerates the number of stopping sets with size i of C with paritycheck matrix H. Denote H ∗ , the paritycheck matrix, consisting of all the nonzero codewords in the dual code C ⊥. In this paper, we study stopping sets and stopping set distributions of some residue algebraic geometry (AG) codes with paritycheck matrix H ∗. First, we give two descriptions of stopping sets of residue AG codes. For the simplest AG codes, i.e., the generalized Reed–Solomon codes, it is easy to determine all the stopping sets. Then, we consider the AG codes from elliptic curves. We use the group structure of rational points of elliptic curves to present a complete characterization of stopping sets. Then, the stopping sets, the stopping set distribution, and the stopping distance of the AG code from an elliptic curve are reduced to the search, counting, and decision versions of the subset sum problem in the group of rational points of the elliptic curve, respectively. Finally, for some special cases, we determine the stopping set distributions of the AG codes from elliptic curves. Index Terms — Algebraic geometry codes, elliptic curves, stopping distance, stopping sets, stopping set distribution, subset sum problem. I.