Abstract Using algebraic techniques, one of the co-authors has designed a [155,64,20] low density parity check code based on permutation matrices. This code is quasi-cyclic by construction and hence admits a convolutional representation. A set of low density parity check convolutional codes is derived from this quasi-cyclic code and its generalizations. We investigate the performance of these c...

A generalization of Sullivan inequality on the ratio of the probability of a linear code to that of any of its cosets is proved. Starting from this inequality, a sufficient condition for successful decoding of linear codes by a probabilistic method is derived. A probabilistic decoding algorithm for “low–density parity–check codes” is also analyzed. The results obtained allow one to estimate exp...

Generalized quasi-cyclic (GQC) codes form a wide and useful class of linear codes that includes thoroughly quasi-cyclic codes, finite geometry (FG) low density parity check (LDPC) codes, and Hermitian codes. Although it is known that the systematic encoding of GQC codes is equivalent to the division algorithm in the theory of Gröbner basis of modules, there has been no algorithm that computes G...

We propose a new decoding algorithm of LDPC codes over the binary erasure channel (BEC). The proposed decoding algorithm is an iterative one which uses the sparseness structure of the parity-check matrix of LDPC codes. We show by simulation results that the proposed decoding algorithm and the conventional decoding algorithm have a trade-off between remaining erasure rate and complexity. Keyword...

In this correspondence, low-density parity-check (LDPC) codes are applied to lossy source coding and we study how the asymptotic performance of MacKay’s LDPC codes depends on the sparsity of the parity-check matrices in the source coding of the binary independent and identically distributed (i.i.d.) source with Pr = 1 = 0 5. In the sequel, it is shown that an LDPC code with column weight (log )...

In this paper, we propose to leverage the simple and explicit parity checks inherent in low-density parity-check (LDPC) codes to realize dominant error events detection without code rate penalty. This is enabled by enforcing a very weak constraint on LDPC code parity check matrix structure. Such a constraint can be readily satisfied by most structured LDPC codes ever studied in the open literat...

An algorithm for encoding low-density parity check (LDPC) codes is investigated. The algorithm computes parity check symbols by solving a set of sparse equations, and the triangular factorization is employed to solve the equations efficiently. It is shown analytically and experimentally that the proposed algorithm is more efficient than the Richardson’s encoding algorithm if the code has a smal...

The main goal of coding theory is to devise efficient systems to exploit the full capacity of a communication channel, thus achieving an arbitrarily small error probability. Low Density Parity Check (LDPC) codes are a family of block codes—characterised by admitting a sparse parity check matrix—with good correction capabilities. In the present paper the orbits of subspaces of a finite projectiv...

The performance of non-binary linear block codes is studied in this paper via the derivation of new upper bounds on the error probability under ML decoding. The transmission of these codes is assumed to take place over a memoryless and symmetric channel. The new bounds, which rely on the Gallager bounding technique, are applied to expurgated ensembles of non-binary and regular low-density parit...

In this paper, I will give a brief introduction to the theory of low-density parity-check codes, and their decoding. I will emphasize the case of correcting erasures as it is still the best understood and most accessible case. At the end of the paper, I will also describe more recent developments.