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Irish Signals and Systems Conference

Construction of girth 8 LDPC codes based on finite lattice geometries

J. Craddock, Mark F. Flanagan, A.D. Fagan

Published Jan 1, 2006• 1 citation
Publisher

Abstract

This paper presents a novel method for constructing low density parity check (LDPC) codes with a girth of up to 8. These codes are based on the structural properties of finite lattice geometries. Results are presented which show that these codes perform well over AWGN channels with iterative decoding. These codes also have several implementation advantages, which are presented.

DOI
10.1049/cp:20060477

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This paper presents a novel method for constructing low density parity check (LDPC) codes with a girth of up to 8. These codes are based on the structural properties of finite lattice geometries. Results are presented which show that these codes perform well over AWGN channels with iterative decoding. These codes also have several implementation advantages, which are presented.
This paper presents a geometric approach to the construction of low-density parity-check (LDPC) codes. Four classes of LDPC codes are constructed based on the lines and points of Euclidean and projective geometries over finite fields. Codes of these four classes have good minimum distances and their Tanner (1981) graphs have girth 6. Finite-geometry LDPC codes can be decoded in various ways, ranging from low to high decoding complexity and from reasonably good to very good performance. They perform very well with iterative decoding. Furthermore, they can be put in either cyclic or quasi-cyclic form. Consequently, their encoding can be achieved in linear time and implemented with simple feedback shift registers. This advantage is not shared by other LDPC codes in general and is important in practice. Finite-geometry LDPC codes can be extended and shortened in various ways to obtain other good LDPC codes. Several techniques of extension and shortening are presented. Long extended finite-geometry LDPC codes have been constructed and they achieve a performance only a few tenths of a decibel away from the Shannon theoretical limit with iterative decoding.
This paper,based on algebra and graph,proposes a method for constructing structured low-density parity-check(LDPC) codes.In this method,the authors design a connected graph with three kinds of special path to ensure that the Tanner graph of the parity check matrix mapped from the connected graph is without short cycles.The construction method results in two classes of(3,k)-regular quasi-cyclic LDPC codes with girth 8 and 12 respectively.Furthermore,by extending this construction method,two classes of LDPC codes with column weight 2,whose girths are respectively 16 and 24,can be obtained.The simulation studies show that these codes can achieve better performance than randomly constructed regular LDPC codes over AWGN channels with iterative decoding.At the BER of 10-5,the new proposed(3,6)-regular(4356,2205) QC-LDPC code performs 1.48dB from the Shannon limit.Moreover,it outperforms the random regular LDPC code of comparable parameters by 0.1dB.For the new proposed structured LDPC codes with column weight 2,they can achieve about 2dB coding gains with respect to the random regular LDPC codes with similar block lengths and code rates.
This article describes the different methods to design regular low density parity-check (LDPC) codes with large girth. In graph terms, this corresponds to designing bipartite undirected regular graphs with large girth. Large girth speeds the convergence of iterative decoding and improves the performance at least in the high SNR range, by slowing down the onsetting of the error floor. We reviewed several existing constructions from exhaustive search to highly structured designs based on Euclidean and projective finite geometries and combinatorial designs. We describe GB and TS LDPC codes and compared the BER performance with large girth to the BER performance of random codes. These studies confirm that in the high SNR regime these codes with high girth exhibit better BER performance. The regularity of the codes provides additional advantages that we did not explore in this article like the simplicity of their hardware implementation and fast encoding.