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International Conference on Advanced Communication Technologies and Networking (CommNet)

Construction of New Finite Geometry Non-Binary LDPC Codes With Reduced Alphabets

Zakaria M’rabet, Fouad Ayoub, Mostafa Belkasmi, Faissal El Bouanani

Published Dec 11, 2023• 1 citation
Publisher

Abstract

The finite Euclidean and Projective geometry codes are the most known among One-Step Majority-Logic Decodable codes that are Low-Density Parity-Check codes as well. Unfortunately, these codes are rare, and we do not have a great diversity when designing communication systems with a fixed code-length and code rate. Regarding the non-binary Euclidian and projective geometric Codes, each code is restricted to a unique alphabet. To offer a bigger range of possible alphabets for each code, we propose new construction by redefining the incidence vector of the lines of Euclidean and projective geometries over finite fields. This approach leads to an enrichment of the class of EG and PG codes. An error performance study of some new codes has been carried out and shows good results.

DOI
10.1109/commnet60167.2023.10365279

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The finite Euclidean and Projective geometry codes are the most known among One-Step Majority-Logic Decodable codes that are Low-Density Parity-Check codes as well. Unfortunately, these codes are rare, and we do not have a great diversity when designing communication systems with a fixed code-length and code rate. Regarding the non-binary Euclidian and projective geometric Codes, each code is restricted to a unique alphabet. To offer a bigger range of possible alphabets for each code, we propose new construction by redefining the incidence vector of the lines of Euclidean and projective geometries over finite fields. This approach leads to an enrichment of the class of EG and PG codes. An error performance study of some new codes has been carried out and shows good results.
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.
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.
In this paper, an improved decoding algorithm for codes that are constructed from finite geometries is introduced. The application of this decoding algorithm to Euclidean geometry (EG) and projective geometry (PG) codes is further discussed. It is shown that these codes can be orthogonalized in less than or equal to three steps. Thus, these codes are majority-logic decodable in no more than three steps. Our results greatly reduce the decoding complexity of EG and PG codes in most cases. They should make these codes very attractive for practical use in error-control systems.
Among the classes of LDPC codes that have been constructed and designed, the only class of LDPC codes that are cyclic is the class of codes constructed based on the incidence vectors of lines of finite geometries, called finite geometry LDPC codes. Cyclic finite geometry LDPC codes are known to have large minimum distances and can provide good error performance with very low error-floors using iterative decoding based on belief propagation. Their cyclic structure allows them to be efficiently and systematically encoded with simple shift-registers in linear time with linear complexity. An obvious question is whether, besides cyclic finite geometry LDPC codes, there are other cyclic codes with large minimum distances that can be efficiently decoded iteratively using channel soft information. In this paper, we present one such class of cyclic codes. Codes in this class are two-step majority-logic decodable and they are also constructed based on finite geometries. Two iterative decoding algorithms are devised for this class of cyclic codes and they provide significant coding gain over the two-step majority-logic decoding of codes in this class.