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Electronics Letters

Deterministic π-rotation low-density parity check codes

Rich Echard, Shih-Chun Chang

Published May 9, 2002• 19 citations
Publisher

Abstract

An ensemble of ‘π-rotation’ low-density parity check codes, extending the result in previously reported work by a different team of authors, is introduced. The key features of these codes are a deterministic matrix based on a single permutation vector defined by three integers, good performance with the proper choice of integers and simple circuit implementation.

DOI
10.1049/el:20020305

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An ensemble of ‘π-rotation’ low-density parity check codes, extending the result in previously reported work by a different team of authors, is introduced. The key features of these codes are a deterministic matrix based on a single permutation vector defined by three integers, good performance with the proper choice of integers and simple circuit implementation.
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.
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.
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.
In this paper we present two theoretical results that lead to the development of good π-rotation LDPC codes. The first theorem derives bounds on the weight of the parity vector when the weight of the information vector is two. The second theorem establishes the expected number of short-loops in the parity check matrix. We then present construction tools that are used to reduce the complexity of searching minimum distance and girth for π-rotation LDPC codes. Several performance examples are given