Some Construction Of Low-Density Parity-Check Codes

In this paper, we develop a deterministic form of low-density parity-check (LDPC) codes. The original LDPC codes were developed in 1962 by Robert Gallager and are based on random parity-check matrix construct, and rediscov-ered 1996 by D.J.C. Mackay that the LDPC codes could be modified slightly to provide the most efficient error correction scheme. These newer LDPC codes, based on an irregular column weight in the underlying matrix, are still defined with random construction techniques.Ever since the rediscovery, LDPC codes have become focal points of research. LDPC codes can achieve near Shannon limit performance, especially for long random or pseudo-random LDPC codes. However, random LDPC codes have draws of high complexity for encoding, it is hard for hardware implement. On the other hand, quasi-cyclic (QC) LDPC codes can be efficiently encoded using simple shift registers with linear complexity.So intense research has been focused in the constructing of QC-LDPC codes. Yu Kou and Shu Lin et al. constructed LDPC codes based on finite geometry and pointed that constructing LDPC codes based on balanced incomplete block designs (BIBD) is a good direction. Based on some special classes of BIBD's that are constructed from prime fields, B. Honary et al. constructed some QC-LDPC codes with good error correcting performance. However, they require the condition of "h odd", for many prime fields GF(p), p is a prime of the form 12t + 1 or 20t + 1, they don't have primitive element a, such that "α4t-1=αh, h odd", say GF(157),GF(193),GF(313). or "α4t + 1=αh, h odd", say GF(181), GF(401), GF(461). For these prime fields, we still can use them to construct QC-LDPC codes through designing special classes of PBIBD's. For prime p of the form 30t+1, we can construct QC-LDPC codes too.In this paper, we construct three classes of special partially balanced incom-plete block designs (PBIBDs) based on finite fields, and use them to construct regular LDPC codes. The constructed codes are free of 4-cycles and with high flexibility for the choosing of the code rate. In particular, they can also be quasi-cyclic. For the first two of them, we loosen the constraints on the fields which B.Honary et al. used to construct QC-LDPC codes. The last one may be new. Simulation results show that some of our codes can have slightly better performance than random regular LDPC codes over AWGN channel.The structure of this paper is as follows. Firstly, we introduce the devel-opment of coding theory. Secondly, we give some basic concepts of linear codes and LDPC codes. Thirdly, we present some decoding algorithms of LDPC codes. Lastly, we give the main results of this paper.