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International Conference on Software, Telecommunications and Computer Networks (SoftCOM)

Sparse generator matrices for some families of Quasi-Cyclic Low-Density Parity-Check codes

Marco Baldi, Giovanni Cancellieri, Franco Chiaraluce

Published Sep 1, 2014• 3 citations
Publisher

Abstract

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.

DOI
10.1109/softcom.2014.7039123

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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 correspondence, the construction of low-density parity-check (LDPC) codes from circulant permutation matrices is investigated. It is shown that such codes cannot have a Tanner graph representation with girth larger than , and a relatively mild necessary and sufficient condition for the code to have a girth of or is derived. These results suggest that families of LDPC codes with such girth values are relatively easy to obtain and, consequently, additional parameters such as the minimum distance or the number of redundant check sums should be considered. To this end, a necessary condition for the codes investigated to reach their maximum possible minimum Hamming distance is proposed.
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.
Generalized low-density parity-check (GLDPC) codes, where single parity-check constraints on the code bits are replaced with generalized constraints (an arbitrary linear code), are a promising class of codes for low-latency communication. The block error rate performance of the GLDPC codes, combined with a complementary outer code, has been shown to outperform a variety of state-of-the-art code and decoder designs with suitable lengths and rates for the 5G ultra-reliable low-latency communication (URLLC) regime. A major drawback of these codes is that it is not known how to construct appropriate polynomial matrices to encode them efficiently. In this paper, we analyze practical constructions of quasi-cyclic GLDPC (QC-GLDPC) codes and show how to construct polynomial generator matrices in various forms using minors of the polynomial matrix. The approach can be applied to fully generalized matrices or partially generalized (with mixed constraint node types) to find better performance/rate trade-offs. The resulting encoding matrices are presented in useful forms that facilitate efficient implementation. The rich substructure displayed also provides us with new methods of determining low weight codewords, providing lower and upper bounds on the minimum distance and often giving those of weight equal to the minimum distance. Based on the minors of the polynomial parity-check matrix, we also give a formula for the rank of any parity-check matrix representing a QC-LDPC or QC-GLDPC code, and hence, the dimension of the code. Finally, we show that by applying double graph-liftings, the code parameters can be improved without affecting the ability to obtain a polynomial generator matrix.
Quasi-cyclic (QC) low-density parity-check (LDPC) codes have several appealing properties regarding decoding, storage requirements and encoding aspects. In this paper, we focus on the QC LDPC codes over GF(q) whose parity-check matrices have fixed column weight j=2. By investigating two subgraphs in the Tanner graphs of the corresponding base matrices, we derive two upper bounds on the minimum Hamming distance for this class of codes. In addition, a method is proposed to construct QC LDPC codes over GF(q), which have good Hamming distance distributions. Simulations show that our designed codes have good performance.
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.