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The π-rotation low-density parity check codes

Rich Echard, Shih-Chun Chang

Published Nov 13, 2002• 51 citations
Publisher

Abstract

We introduce an ensemble of quasi-regular low-density parity check error control codes we call /spl pi/-rotation LDPC codes. These codes are completely defined by a single permutation vector of length n/8 where n is the length of the code. We present an efficient encoding scheme and circuit design based on the single permutation. Simulation results indicate that these codes perform as well as regular low density parity check codes based on completely random matrix constructions. These features make the /spl pi/-rotation code valuable for practical communication system implementation.

DOI
10.1109/glocom.2001.965564

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We introduce an ensemble of quasi-regular low-density parity check error control codes we call /spl pi/-rotation LDPC codes. These codes are completely defined by a single permutation vector of length n/8 where n is the length of the code. We present an efficient encoding scheme and circuit design based on the single permutation. Simulation results indicate that these codes perform as well as regular low density parity check codes based on completely random matrix constructions. These features make the /spl pi/-rotation code valuable for practical communication system implementation.
π-rotation LDPC codes are characterized by their simple encoding structure,little memory requirement and easy circuit implementation.A new regular encoding structure for π-rotation LDPC codes is provided,whose parity check matrix His described by a few indexes and the number of which is less than the code length.This encoding structure is also easy to adjust code rate and code length.A new simple but efficient encoding scheme is also given to encode codeword with the aid of the indexes.Two-dimension arrays are used to store the belief information between check nodes and variable nodes,and then the BP Algorithm is developed.Simulation results indicate that these codes perform as well as regular low-density parity check codes based on completely random matrix constructions.
The original low-density parity-check (LDPC) codes were developed by Robert Gallager in early 1960 and are based on a random parity-check matrix construction. In the mid 1990's it was discovered that LDPC codes could be modified slightly to provide the more powerful error correction. These newer LDPC codes, based on an irregular column weight in the underlying check matrix, were still defined with random construction techniques. The Pi-rotation LDPC codes discovered by Echard are a family of LDPC codes completely defined by a small set of integers and have several symmetrical features that are exploited to build efficient encoding and decoding designs. The Pi-rotation codes can be extended to include irregular matrix patterns to obtain the highest performance. In this dissertation we develop a heuristic algorithm to find the best parity-check matrix for Pi-rotation LDPC codes.
In this correspondence we present a special class of quasi-cyclic low-density parity-check (QC-LDPC) codes, called block-type LDPC (B-LDPC) codes, which have an efficient encoding algorithm due to the simple structure of their parity-check matrices. Since the parity-check matrix of a QC-LDPC code consists of circulant permutation matrices or the zero matrix, the required memory for storing it can be significantly reduced, as compared with randomly constructed LDPC codes. We show that the girth of a QC-LDPC code is upper-bounded by a certain number which is determined by the positions of circulant permutation matrices. The B-LDPC codes are constructed as irregular QC-LDPC codes with parity-check matrices of an almost lower triangular form so that they have an efficient encoding algorithm, good noise threshold, and low error floor. Their encoding complexity is linearly scaled regardless of the size of circulant permutation matrices.
In this study, a new method for constructing low-density parity-check (LDPC) codes is presented. This construction is based on permutation matrices which come from a finite abstract group and hence the codes constructed in this manner are called group permutation low-density parity-check (GP-LDPC) codes. A necessary and sufficient condition under which a GP-LDPC code has a cycle is given and some properties of these codes are investigated. A class of flexible-rate GP-LDPC codes without cycles of length four is also introduced. Simulation results show that GP-LDPC codes perform very well with the iterative decoding and can outperform their random-like counterparts.
Cyclically-coupled quasi-cyclic low-density parity-check (CC-QC-LDPC) codes are a new class of QC-LDPC codes which can achieve excellent error performance and relatively low hardware requirement. In this paper, we modify the CC-QC-LDPC code construction by using random permutation matrices instead of circulant permutation matrices, forming the “random-permutation-matrix-based CC-LDPC (RP-CC-LDPC) code”. The objective is to achieve a better error performance under the same code length. We simulate the bit error rate using FPGA simulations. We also compare the BER results and decoder complexity of the above codes with those of regular and irregular QC-LDPC codes under the same code length and code rate.