On Decoding of Low-Density Parity-Check Codes Over the Binary Erasure Channel

Abstract

Abstract-This paper investigates decoding of low-density parity-check (LDPC) codes over the binary erasure channel (BEC). We study the iterative and maximum-likelihood (ML) decoding of LDPC codes on this channel. We derive bounds on the ML decoding of LDPC codes on the BEC. We then present an improved decoding algorithm. The proposed algorithm has almost the same complexity as the standard iterative decoding. However, it has better performance. Simulations show that we can decrease the error rate by several orders of magnitude using the proposed algorithm. We also provide some graph-theoretic properties of different decoding algorithms of LDPC codes over the BEC which we think are useful to better understand the LDPC decoding methods, in particular, for finite-length codes.

On Achievable Rates and Complexity of LDPC Codes Over Parallel Channels: Bounds and Applications

“…1; it is shown in this figure that results from [1], [9], [10], [12], [14], [17], which correspond to the achievable rates and decoding complexity of LDPC codes (with and without puncturing), follow as special cases of the bounds introduced in this paper. who pointed out Remarks 3.1 and 4.2 in this paper.…”

“…The latter result is used for the derivation of lower bounds on the decoding complexity of randomly and intentionally punctured LDPC codes for MBIOS channels; looser versions of these bounds suggest a simplified re-derivation of previously reported bounds on the decoding complexity of randomly punctured LDPC codes (as shown in the appendix). Finally, Section VI summarizes our discussion, and presents a diagram which shows interconnections between the theorems introduced in this paper and some other previously reported results from [1], [9], [10], [12], [14], [17]. The preliminary material on ensembles of LDPC codes and notation required for this paper are introduced in [13] and [17,Section 2].…”

Erasures and Error-Correcting Codes

“…There are a total of n − k parity check equations and provided the erased bit positions correspond to independent columns of the H matrix, each of the erased bits may be solved using a parity-check equation derived by the classic technique of Gaussian reduction [15][16][17]. For maximum distance separable (MDS) codes, [15], any set of s erasures are correctable by the code provided that…”

“…It is straightforward, by computer simulation, to evaluate the erasure correcting performance of the code by generating a pattern of erasures randomly and solving these in turn using the parity-check equations. This procedure corresponds to maximum likelihood (ML) decoding [6,17]. Moreover, the codeword responsible for any instances of non-MDS performance, (due to this erasure pattern) can be determined by back substitution into the solved parity-check equations.…”

“…Furthermore, the erasure performance of LDPC codes, in particular, has been used as a measure of predicting the code performance for the additive white Gaussian noise (AWGN) channel [6,17]. One of the first analyses of the erasure correction performance of particular linear block codes is provided in a key-note paper by Dumer and Farrell [7] who derive the erasure correcting performance of long binary BCH codes and their dual codes.…”