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IEEE Communications Letters

Design considerations leading to the development of good π-rotation LDPC codes

Rich Echard, Shih-Chun Chang

Published May 1, 2005• 13 citations
Publisher

Abstract

In this letter we present two theoretical results that lead to the development of good /spl pi/-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. Several performance examples are given.

DOI
10.1109/lcomm.2005.1431167

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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.

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In this letter we present two theoretical results that lead to the development of good /spl pi/-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. Several performance examples are given.
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
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.
The existing design of quasi-cyclic(QC) low-density parity-check(LDPC) codes has not considered the problems of short length girths and the rows' dependency.The first problem leads to the BER performance of QC LDPC codes to be much poorer than that of randomly constructed LDPC codes,while the second problem leads to the difficulty of construction of generator matrices from parity-check matrices.To solve the first problem,the constraint conditions for designing the QC LDPC codes are proposed,and the dimension and shift factors of the circulant matrices of the given sparse parity-check matrix are adjusted according to the test results of both girth 4 and girth 6.To solve the second problem,an approach of irregular QC LDPC codes is proposed.The proposed approach is to replace some sub-matrices by zero matrices and identity matrices at special positions in the given sparse parity-check matrix so as to get a nonsingular square matrix for the construction of the generator matrix.Though adopting bidiagonal submatrices in parity-check matrix can solve the rows' dependency problem,many code words with low weights will occur,which leads to the BER performance can not be better by increasing code lengths.Examples are provided for the proposed design of QC LDPC codes,and computer simulation results show that the proposed QC LDPC codes achieve good BER performance.
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.