Comparison of constructions of irregular Gallager codes

Abstract

The low density parity check codes whose performance is closest to the Shannon limit are`Gallager codes' based on irregular graphs. We compare alternative methods for constructing these graphs and present two results. First, we nd a`super{Poisson' construction which gives a small improvement in empirical performance over a random construction. Second, whereas Gallager codes normally take N 2 time to encode, we investigate constructions of regular and irregular Gallager codes which allow more rapid encoding and have smaller memory requirements in the encoder. We nd that thesè fast{encoding' Gallager codes have equally good performance.

Efficient erasure correcting codes

“…4 This paper appears to have been influential. First, the idea of using irregular codes was taken up and extended by other researchers (see, e.g., [14]). Second, the main "concentration theorem" of [10] was extended to a large class of channel models in a landmark paper by Richardson and Urbanke [22], which first appeared in 1998.…”

Low-Density Parity-Check Codes and Their Rateless Relatives

“…However, in both cases, the performance exhibited by the resultant codes based on cascaded graphs appeared to be inferior to that of standard LDPC codes 4 since clearly, the block length of each stage of the cascaded code is lower than that of the overall length of the standard LDPC code. MacKay et al in [80] suggested that the parity-check matrix must be constrained to be in an approximate lower triangular (ALT) form depicted in Figure 4 which guarantees a linear increase of the encoding complexity. Richardson and Urbanke in [81] proved that in general, the encoding complexity increases nearly linear with the block length, being quadratic only in a small term g 2 , where g is referred to as the gap [82], which is a measure of the 'distance' [82] between the PCM and the lower triangular matrix as shown in Figure 4.…”

Performance of low density parity check coded continuous phase frequency shift keying (LDPCC‐CPFSK) over fading channels

“…Encoder complexity of LDPC codes is proportional with the square of code length (n 2 ), therefore, encoding process time is longer than turbo encoding. In construction of LDPC codes, choosing appropriate upper triangle parity check matrix, provides significant encoding complexity reduction as in References [7,8]. These kind of matrix transformations may converse the sparsity of the original matrix, but do not affect code performance.…”