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Computer Knowledge and Technology

Non-Binary LDPC Codes Design Based on Euclidean Geometries

Guihua Zeng

Published Jan 1, 2010• 2 citations
OpenAlex

Abstract

This paper presents a Euclidean geometric approach to the construction of Non-Binary low-density parity-check(LDPC) codes.Codes of this class have girth at least 6.They perform very well with iterative decoding.Furthermore,these codes can be decoded with simple shift registers,with linear complexity and decoded in the semi-parallel way.The simulation results show that their performance is much better than the binary LDPC codes.

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This paper presents a Euclidean geometric approach to the construction of Non-Binary low-density parity-check(LDPC) codes.Codes of this class have girth at least 6.They perform very well with iterative decoding.Furthermore,these codes can be decoded with simple shift registers,with linear complexity and decoded in the semi-parallel way.The simulation results show that their performance is much better than the binary LDPC codes.
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.
A new method for constructing quasi-cyclic (QC) low-density parity-check (LDPC) codes based on the structural properties of Euclidean geometries is presented. The proposed method changes the incidence vectors of some lines in the original Euclidean geometry method to construct much more parity check matrices and results in a large class of QC-LDPC codes with girth of at least 6. Simulations show that some codes in this class have better performance and lower error floor than the original Euclidean geometry codes.
Low density parity check (LDPC) codes with iterative decoding based on belief propagation achieve astonishing error performance close to Shannon limit. No algebraic or geometric method for constructing these codes has been reported and they are largely generated by computer search. As a result, encoding of long LDPC codes is in general very complex. This paper presents two classes of high rate LDPC codes whose constructions are based on finite Euclidean and projective geometries, respectively. These classes of codes a.re cyclic and have good constraint parameters and minimum distances. Cyclic structure adows the use of linear feedback shift registers for encoding. These finite geometry LDPC codes achieve very good error performance with either soft-decision iterative decoding based on belief propagation or Gallager's hard-decision bit flipping algorithm. These codes can be punctured or extended to obtain other good LDPC codes. A generalization of these codes is also presented.
A construction technique is proposed for low-density parity-check (LDPC) codes based on finite Euclidean geometries EG(m,2s). These codes are shown to be regular Gallager codes with Tanner graphs of girth eight. The minimum distance of these codes is shown to be lower-bounded by 2m. The codes are also amenable to an efficient partly parallel decoder implementation, which may be used in conjunction with the turbo decoding message passing (TDMP) algorithm for LDPC decoding. Finally, simulation results show that these codes have very good error-correcting performance