We investigate thestopping redundancy hierarchy oflinear blockcodesanditsconnection topermutation de- coding techniques. An element intheordered list ofstopping redundancy values represents thesmallest numberofpossibly linearly dependent rowsinanyparity-check matrix ofacode thatavoids stopping setsofup toa givensize. Redundant parity-check equations canbeshowntohavea similar effect ondecoding performance aspermuting thecoordinates ofthe received codeword according toaselected setofautomorphisms ofthecode.Basedonthisfinding we develop newdecoding strategies fordatatransmission overthebinary erasure channel thatcombine iterative message passing andpermutation decoding inordertoavoiderrors confined tostopping sets. We also introduce thenotion ofs-SADsets, containing thesmallest numberofautomorphisms ofacodewiththeproperty thatthey moveanysetofnotmorethanserasures intopositions that donotcorrespond tostopping setswithin ajudiciously chosen parity-check matrix.
We investigate the stopping redundancy hierarchy of linear block codes and its connection to permutation decoding techniques. An element in the ordered list of stopping redundancy values represents the smallest number of possibly linearly dependent rows in any parity-check matrix of a code that avoids stopping sets of up to a given size. Redundant parity-check equations can be shown to have a similar effect on decoding performance as permuting the coordinates of the received codeword according to a selected set of automorphisms of the code. Based on this finding we develop new decoding strategies for data transmission over the binary erasure channel that combine iterative message passing and permutation decoding in order to avoid errors confined to stopping sets. We also introduce the notion of s-SAD sets, containing the smallest number of automorphisms of a code with the property that they move any set of not more than s erasures into positions that do not correspond to stopping sets within a judiciously chosen parity-check matrix.
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 Lovasz's local lemma (LLL) and Bonferroni-type inequalities, and specialize them for codes with cyclic parity-check matrices. Based on the observed properties of parity-check 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 (ML) decoding.
Abstract — We introduce the notion of the stopping redundancy hierarchy of a linear block code as a measure of the trade-off 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 Bonferroni-type inequalities, and specialize them for codes with cyclic parity-check matrices. Based on the observed prop-erties of parity-check 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.
We extend the framework for studying the stopping redundancy of a linear block code by introducing and analyzing the stopping redundancy hierarchy.The stopping redundancy hierarchy of a code represents a measure of the trade-off between performance and complexity of iteratively decoding a code used over the binary erasure channel.It is defined as an ordered list of positive integers in which the ith entry, termed the i-th stopping redundancy, corresponds to the minimum number of rows in any parity-check matrix of the code that has stopping distance at least i.In particular, we derive lower and upper bounds for the i-th stopping redundancy of a code by using probabilistic methods and Bonferroni-type inequalities.Furthermore, we specialize the findings for cyclic codes, and show that parity-check matrices in cyclic form have some desirable redundancy properties.We also propose to investigate the influence of the generator codeword of the cyclic parity-check matrix on its stopping distance properties.
Stopping sets play a crucial role in failure events of iterative decoders over a binary erasure channel (BEC). The ℓth stopping redundancy is the minimum number of rows in the parity-check matrix of a code, which contains no stopping sets of size up to ℓ. In this paper, a notion of coverable stopping sets is defined. In order to achieve maximum-likelihood performance under iterative decoding over the BEC, the parity-check matrix should contain no coverable stopping sets of size ℓ, for 1 ≤ ℓ ≤ n-k, where n is the code length, k is the code dimension. By estimating the number of coverable stopping sets, we obtain upper bounds on the ℓth stopping redundancy, 1 ≤ ℓ ≤ n-k. The bounds are derived for both specific codes and code ensembles. In the range 1 ≤ ℓ ≤ d-1, for specific codes, the new bounds improve on the results in the literature. Numerical calculations are also presented.