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IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences

Some Classes of Quasi-Cyclic LDPC Codes: Properties and Efficient Encoding Method

Hideaki Fujita

Published Dec 1, 2005• 27 citations
Publisher

Abstract

Low-density parity-check (LDPC) codes are one of the most promising next-generation error-correcting codes. For practical use, efficient methods for encoding of LDPC codes are needed and have to be studied. However, it seems that no general encoding methods suitable for hardware implementation have been proposed so far and for randomly constructed LDPC codes there have been no other methods than the simple one using generator matrices. In this paper we show that some classes of quasi-cyclic LDPC codes based on circulant permutation matrices, specifically LDPC codes based on array codes and a special class of Sridhara-Fuja-Tanner codes and Fossorier codes can be encoded by division circuits as cyclic codes, which are very easy to implement. We also show some properties of these codes.

DOI
10.1093/ietfec/e88-a.12.3627

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Low-density parity-check (LDPC) codes are one of the most promising next-generation error-correcting codes. For practical use, efficient methods for encoding of LDPC codes are needed and have to be studied. However, it seems that no general encoding methods suitable for hardware implementation have been proposed so far and for randomly constructed LDPC codes there have been no other methods than the simple one using generator matrices. In this paper we show that some classes of quasi-cyclic LDPC codes based on circulant permutation matrices, specifically LDPC codes based on array codes and a special class of Sridhara-Fuja-Tanner codes and Fossorier codes can be encoded by division circuits as cyclic codes, which are very easy to implement. We also show some properties of these codes.
In this correspondence we present a special class of quasi-cyclic low-density parity-check (QC-LDPC) codes, called block-type LDPC (B-LDPC) codes, which have an efficient encoding algorithm due to the simple structure of their parity-check matrices. Since the parity-check matrix of a QC-LDPC code consists of circulant permutation matrices or the zero matrix, the required memory for storing it can be significantly reduced, as compared with randomly constructed LDPC codes. We show that the girth of a QC-LDPC code is upper-bounded by a certain number which is determined by the positions of circulant permutation matrices. The B-LDPC codes are constructed as irregular QC-LDPC codes with parity-check matrices of an almost lower triangular form so that they have an efficient encoding algorithm, good noise threshold, and low error floor. Their encoding complexity is linearly scaled regardless of the size of circulant permutation matrices.
Cyclically-coupled quasi-cyclic low-density parity-check (CC-QC-LDPC) codes are a new class of QC-LDPC codes which can achieve excellent error performance and relatively low hardware requirement. In this paper, we modify the CC-QC-LDPC code construction by using random permutation matrices instead of circulant permutation matrices, forming the “random-permutation-matrix-based CC-LDPC (RP-CC-LDPC) code”. The objective is to achieve a better error performance under the same code length. We simulate the bit error rate using FPGA simulations. We also compare the BER results and decoder complexity of the above codes with those of regular and irregular QC-LDPC codes under the same code length and code rate.
Low-density parity-check (LDPC) codes based on affine permutation matrices, APM-LDPC codes, have been attracted recently, because of some advantages rather than QC-LDPC codes in minimum-distance, cycle distribution and error-rate performance. In this letter, circulant and anti-circulant permutation matrices are used to define a class of LDPC codes, called AQC-LDPC codes, which can be considered as an special case of APM-LDPC codes. In fact, each AQC-LDPC code can be verified by a sign matrix and a slope matrix which are helpful to show each cycle in the Tanner graph by a modular linear equation. For the normal sign matrix A , if -1 ϵ, A, it is shown that the corresponding AQC-LDPC code has maximum-girth 8. Finally, two explicit constructions for AQC-LDPC codes with girths 6, 8 are presented which have some benefits rather than the explicitly constructed QC and APM LDPC codes in minimum-distance, cycle distributions and bit-error-rate performances.
An approach for studying quasi-cyclic codes based on matrix transformations via Fourier transforms and row and column permutations is presented. These transformations put a parity-check matrix in the form of an array of circulant matrices into a diagonal array of matrices of the same size over an extension field. The approach is used to characterize certain structural properties of low-density parity-check (LDPC) codes such as the girths of their Tanner graphs. Many constructions of quasi-cyclic LDPC codes can be unified under the proposed approach.
In this paper, we proposed a novel method for constructing quasi-cyclic low-density parity-check (QC-LDPC) codes based on circulant permutation matrices via a simple quadratic congruential equation. The main advantage is that QC-LDPC codes with a variety of block lengths and rates can be easily constructed with no cycles of length four or less. Simulation results show that the proposed QC-LDPC codes perform slight better than the random regular LDPC codes for short to moderate block lengths and have almost the same performance as Sridara-Fuja-Tanner codes.
In this paper, we proposed a new method for constructing quasi-cyclic low-density parity-check (QC-LDPC) codes based on circulant permutation matrices via the one-coincidence sequences. The main advantage is that QC-LDPC codes with a variety of block lengths and rates can be easily constructed with no cycles of length four or less. Simulation results show that the proposed QC-LDPC codes perform slight better than the random regular LDPC codes for short to moderate block lengths and have almost the same performance as Sridara- Fuja-Tanner codes.