Construction of Near-Capacity Protograph LDPC Code Sequences With Block-Error Thresholds

Abstract

Density evolution for protograph Low-Density Parity-Check (LDPC) codes is considered, and it is shown that the message-error rate falls double-exponentially with iterations whenever the degree-2 subgraph of the protograph is cyclefree and noise level is below threshold. Conditions for stability of protograph density evolution are established and related to the structure of the protograph. Using large-girth graphs, sequences of protograph LDPC codes with block-error threshold equal to bit-error threshold and block-error rate falling nearexponentially with blocklength are constructed deterministically. Small-sized protographs are optimized to obtain thresholds near capacity for binary erasure and binary-input Gaussian channels.

Protograph LDPC Codes With Block Thresholds: Extension to Degree-One and Generalized Nodes

“…This can be proved by contradiction. For details of the proof, we refer readers to [6,Theorem 1]. So, we have shown that…”

“…Block-error threshold of protograph ensemble is defined as the supremum of the set of ǫ for which probability of block error, denoted by P B , tends to zero as the number of iterations tends to infinity. In [6], sufficient conditions for block-error threshold being equal to bit-error threshold have been derived using the following two steps:…”

“…Protographs corresponding to rate-1/5 and rate-1/3 have 10 and 22 information nodes, respectively. Base matrices of RED(G) corresponding to rate-1/5 and rate-1/3 are given in (6) and (7), respectively. Let V RED(G) denote the set of variable nodes in RED(G).…”

“…The block-error threshold condition for BIAWGN channel is same as block-error threshold condition for BEC and can be proved using a Bhattacharya parameter argument as in [6,Theorem 3] and [19,Theorem 2]. Readers interested in designing protographs with block-error threshold can skip the following sections and move to Section IV.…”

Protograph Based Low-Density Parity-Check Codes Design With Mixed Integer Linear Programming

“…Density evolution (DE) and protograph EXIT (PEXIT) techniques are known to facilitate protograph design, typically resulting in the Multi Edge Type (MET) protographs, in which parallel edges are allowed [5]- [10]. MET protograph ensembles have been shown to achieve capacityapproaching thresholds [5], with DE-based optimization enabling to design small-sized protographs, for which the derived LDPC codes are asymptotically close to the capacity for BEC and BIAWGN channels [6]. Numerous other work focus on applications for other channels and specific processing models.…”