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IEEE Information Theory Workshop (ITW)

Performance and complexity of ψ-unitary QC-LDGM codes

Marco Baldi, Franco Chiaraluce

Published Jan 1, 2009• 0 citations
Publisher

Abstract

In this paper we assess, through theoretical arguments and numerical simulations, the performance and complexity of a novel family of quasi-cyclic low-density generator matrix (QC-LDGM) codes. The design of such codes, denoted as ψ-unitary codes, is based on a class of binary circulant matrices recently introduced by the authors. Such matrices can have sparse inverse though being neither identity nor permutation matrices. For this reason, ψ-unitary codes can be LDGM codes while avoiding the penalization in minimum distance due to the usage of identity blocks, as occurs in classic LDGM codes. So, they can represent a trade-off in terms of performance and complexity between classic LDGM codes and non-LDGM codes.

DOI
10.1109/itw.2009.5351180

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In this paper we assess, through theoretical arguments and numerical simulations, the performance and complexity of a novel family of quasi-cyclic low-density generator matrix (QC-LDGM) codes. The design of such codes, denoted as ψ-unitary codes, is based on a class of binary circulant matrices recently introduced by the authors. Such matrices can have sparse inverse though being neither identity nor permutation matrices. For this reason, ψ-unitary codes can be LDGM codes while avoiding the penalization in minimum distance due to the usage of identity blocks, as occurs in classic LDGM codes. So, they can represent a trade-off in terms of performance and complexity between classic LDGM codes and non-LDGM codes.
We present a new class of sparse and easily invertible circulant matrices that can have a sparse inverse though not being permutation matrices. Their study is useful in the design of quasi-cyclic low-density generator matrix codes that are able to join the inner structure of quasi-cyclic codes with sparse generator matrices, so limiting the number of elementary operations needed for encoding. Circulant matrices of the proposed class permit to hit both targets without resorting to identity or permutation matrices that may penalize the code minimum distance and often cause significant error floors.
In this paper, we provide a general form for sparse generator matrices of several families of Quasi-Cyclic Low-Density Parity-Check codes. Codes of this kind have a prominent role in literature and applications due to their ability to achieve excellent performance with limited complexity. While some properties of these codes (like the girth length in their associated Tanner graphs) are well investigated, estimating their minimum distance is still an open problem. By obtaining sparse generator matrices for several families of these codes, we prove that they are also Quasi-Cyclic Low-Density Generator Matrix codes, which is an important feature to reduce the encoding complexity, and provides a useful tool for the investigation of their minimum distance.
In this paper a new low complexity method for constructing binary quasi-cyclic low-density parity-check (QC-LDPC) codes is introduced. In the proposed method, each block-column of the parity check matrix H is made by a circulant matrix in a way that the associated Tanner graph is free of cycle four. Each circulant matrix in H is made by a generator column. The generator columns should be selected in a way that each associated circulant matrix and every two distinct circulant matrices are free of cycle four. The generator columns are made by row distance sets. An algorithm for generating distance sets and obtaining circulant matrices with columns of weight three is presented separately. Simplicity of construction and having a good flexible family of quasi cyclic LDPC codes both in rate and length are the main properties of the proposed method. The performance of the proposed codes is compared with that of the random-like and Array LDPC codes over an AWGN channel. Simulation results show that from the performance perspective, the constructed codes are competitive with random-like and Array LDPC codes.
The existing design of quasi-cyclic(QC) low-density parity-check(LDPC) codes has not considered the problems of short length girths and the rows' dependency.The first problem leads to the BER performance of QC LDPC codes to be much poorer than that of randomly constructed LDPC codes,while the second problem leads to the difficulty of construction of generator matrices from parity-check matrices.To solve the first problem,the constraint conditions for designing the QC LDPC codes are proposed,and the dimension and shift factors of the circulant matrices of the given sparse parity-check matrix are adjusted according to the test results of both girth 4 and girth 6.To solve the second problem,an approach of irregular QC LDPC codes is proposed.The proposed approach is to replace some sub-matrices by zero matrices and identity matrices at special positions in the given sparse parity-check matrix so as to get a nonsingular square matrix for the construction of the generator matrix.Though adopting bidiagonal submatrices in parity-check matrix can solve the rows' dependency problem,many code words with low weights will occur,which leads to the BER performance can not be better by increasing code lengths.Examples are provided for the proposed design of QC LDPC codes,and computer simulation results show that the proposed QC LDPC codes achieve good BER performance.
A class of algebraically structured quasi-cyclic (QC) low-density parity-check (LDPC) codes and their convolutional counterparts is presented. The QC codes are described by sparse parity-check matrices comprised of blocks of circulant matrices. The sparse parity-check representation allows for practical graph-based iterative message-passing decoding. Based on the algebraic structure, bounds on the girth and minimum distance of the codes are found, and several possible encoding techniques are described. The performance of the QC LDPC block codes compares favorably with that of randomly constructed LDPC codes for short to moderate block lengths. The performance of the LDPC convolutional codes is superior to that of the QC codes on which they are based; this performance is the limiting performance obtained by increasing the circulant size of the base QC code. Finally, a continuous decoding procedure for the LDPC convolutional codes is described.