Linear Time Encoding of LDPC Codes

Abstract

Abstract-In this paper, we propose a linear complexity encoding method for arbitrary LDPC codes. We start from a simple graph-based encoding method "label-and-decide." We prove that the "label-and-decide" method is applicable to Tanner graphs with a hierarchical structure-pseudo-trees-and that the resulting encoding complexity is linear with the code block length. Next, we define a second type of Tanner graphs-the encoding stopping set. The encoding stopping set is encoded in linear complexity by a revised label-and-decide algorithm-the "label-decide-recompute." Finally, we prove that any Tanner graph can be partitioned into encoding stopping sets and pseudo-trees. By encoding each encoding stopping set or pseudo-tree sequentially, we develop a linear complexity encoding method for general low-density parity-check (LDPC) codes where the encoding complexity is proved to be less than 4 1 M 1 ( k 0 1), where M is the number of independent rows in the parity-check matrix and k represents the mean row weight of the parity-check matrix.

Improved progressive edge-growth algorithm for fast encodable LDPC codes

“…It is stated [12] (Corollary 4) that if a parity-check matrix H can be transformed into an upper or lower triangular matrix by row and column permutations, the corresponding LDPC code can be encoded in linear number of steps. Obviously, the PEG algorithm can also be tailored to construct a parity-check matrix H having upper triangular structure.…”

“…Condition (A) guarantees that the corresponding codes are linear time encodable [12]. Condition (B) guarantees that the r i parity-check equations in submatrix H i can be used to generate r i parity bits simultaneously [10,13,14] while condition (C) is a necessary condition for condition (B).…”

“…It was proved that the Tanner graph of an upper or lower triangular parity-check matrix could be equivalently transformed into a pseudo-tree and the corresponding LDPC codes could also be encoded in a linear number of steps by the label-and-decide algorithm [12].…”

“…The existence of degree-1 symbol nodes is a necessary condition for a linearencoding algorithm such as the label-and-decide algorithm and the LPEG algorithm [5,12]. However, it was stated that the outbound extrinsic messages of degree-1 nodes would not be updated during the iterative decoding process [16,17].…”

Block-triangularization of parity check matrices for efficient encoding of linear codes

“…In order to overcome the imbalance in encoding and decoding complexity, a considerable number of researches have investigated efficient encoding of LDPC codes [1,5]. Recently, Lu et al [3] succeeded to develop a linear time encoding method for binary LDPC codes.…”