Low-density parity-check codes based on finite geometries: a rediscovery and new results

Abstract

Abstract-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 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.

LDPC Codes

“…Note that all codes with code rate less than 0.25 are excluded from the table and codes of longer lengths may also be constructed. We can also see that some of the codes in Table 12.1 have the same parameters as the Euclidean and projective geometry codes, which have been shown by Jin et al [16] to perform well under iterative decoding. …”

“…The iterative soft decision decoder offers significant improvement over the conventional hard decision majority-logic decoder. Another class of algebraic codes is the class of the Euclidean and projective geometry codes which are discussed in detail by Kou et al [16]. Other algebraic constructions include those that use combinatorial techniques [13][14][15]35].…”

“…12.1a. It is a [16,4,4] code with a λ of 3 and a ρ of 4. The parity-check matrix of the (λ, ρ) Gallager codes always have a fixed number of non-zeros per column and per row, and because of this property, this class of LDPC codes is termed regular LDPC codes.…”

A Class of Low-Density Parity-Check Codes Constructed Based on Reed-Solomon Codes with Two Information Symbols

“…After being overlooked for almost 35 years, this class of codes has been recently rediscovered and shown to form a class of Shannon limit approaching codes [2]- [8]. This class of codes decoded with iterative decoding, such as the sum-product algorithm (SPA) [1], [4]- [6], performs amazingly well.…”

Small weight codewords in the LDPC codes arising from linear representations of geometries

“…In 2001, Kou et al [14] examined classes of LDPC codes defined by incidence structures in finite geometries. Since then, other LDPC codes have been produced based on various incidence structures in discrete mathematics and finite geometry (for example, [8], [9], [10], [17], [28], [29]).…”