We introduce a new method for decoding short and moderate-length linear block codes with dense parity check matrix representations of cyclic form. This approach is termed multiple-bases belief-propagation. The proposed iterative scheme makes use of the fact that a code has many structurally diverse parity-check matrices, capable of detecting different error patterns. We show that this inherent code property leads to decoding algorithms with significantly better performance when compared to standard belief-propagation decoding. Furthermore, we describe how to choose sets of parity-check matrices of cyclic form amenable for multiple-bases decoding, based on analytical studies performed for the binary erasure channel. For several cyclic and extended cyclic codes, the multiple-bases belief propagation decoding performance can be shown to closely follow that of the maximum-likelihood decoder.

We generalize the notion of the stopping redundancy in order to study the smallest size of a trapping set in Tanner graphs of linear block codes. In this context, we introduce the notion of the trapping redundancy of a code, which quantifies the relationship between the number of redundant rows in any parity-check matrix of a given code and the size of its smallest trapping set. Trapping sets with certain parameter sizes are known to cause error-floors in the performance curves of iterative belief propagation (BP) decoders, and it is therefore important to identify decoding matrices that avoid such sets. Bounds on the trapping redundancy are obtained using probabilistic and constructive methods, and the analysis covers both general and elementary trapping sets. Numerical values for these bounds are computed for the [2640, 1320] Margulis code and the class of projective geometry codes, and compared with some new code-specific trapping set size estimates.

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.

A novel soft-decoding method for algebraic block codes is presented. The algorithm is designed for soft-decision decoding and is based on belief-propagation (BP) decoding using multiple bases of the dual code. Compared to other approaches for high-performance BP decoding, this method is conceptually simple and does not change at each stage of the decoding process. With its multiple BP decoders the proposed scheme achieves the performance of a standard BP algorithm with a significantly lower number of iterations per decoder realization. By this means the data delay introduced by decoding is reduced. Moreover, a significant improvement in decoding performance is achieved while keeping the data delay small. It is shown that for selected codes the proposed scheme approaches near maximum likelihood (ML) performance for very small data processing delays.

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.

It is well known that certain combinatorial structures in the Tanner graph of a low-density parity-check (LDPC) code exhibit a strong influence on its performance under iterative decoding. These structures include cycles, stopping/trapping sets, and parameters such as the diameter of the code. In general, it is very hard to find a complete characterization of such configurations in an arbitrary code, and even harder to understand the intricate relationships that exist between these entities. It is, therefore, of interest to identify a simple setting in which all the described combinatorial structures can be enumerated and studied within a joint framework. One such setting is developed in this paper, for the purpose of analyzing the distribution of short cycles and the structure of stopping and trapping sets in Tanner graphs of LDPC codes based on idempotent and symmetric Latin squares. The parity-check matrices of LDPC codes based on Latin squares have a special form that allows for connecting combinatorial parameters of the codes with the number of certain subrectangles in the Latin squares. Subrectangles of interest can be easily identified, and in certain instances, completely enumerated. This study can be extended in several different directions, one of which is concerned with modifying the code design process in order to eliminate or reduce the number of configurations bearing a negative influence on the performance of the code. Another application of the results includes determining to which extent a configuration governs the behavior of the bit-error rate curve in the waterfall and error-floor regions

We analyze the effect of redundant parity-check equations on the error-floor performance of low-density parity- check (LDPC) codes used over the additive white Gaussian noise (AWGN) channel. Our findings show that a large number of iterative decoding errors in the [2640,1320] Margulis code, confined to point trapping sets in the standard Tanner graph, can be corrected if only one redundant parity-check equation is added to the decoder's matrix. We also derive an analytic expression relating the number of rows in the parity-check matrix of a code and the parameters of trapping sets in the code's graph.

In this paper, we introduce a family of structured low-density parity-check (LDPC) codes based on a class of idempotent, symmetric Latin and modified Latin squares. The parity-check matrices of the codes have a block structure with permutation blocks which insures that both their corresponding girth and minimum distance are at least equal to six. The storage requirement for codes from this class is reduced to only one parameter. We also propose a structured method for shortening the codes by removing columns that break a large number of cycles of length six. The storage requirements for the codes obtained using the described procedure consist in memorizing only two integers, while their performance under iterative decoding matches that of random-like codes of comparable length.