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arXiv:1309.1286 · IEEE Transactions on Information Theory

On a Family of Circulant Matrices for Quasi-Cyclic Low-Density Generator Matrix Codes

Marco Baldi, Federico Bambozzi, Franco Chiaraluce

Published Sep 1, 2011• 38 citations
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Abstract

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.

DOI
10.1109/tit.2011.2161953

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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.
This paper presents a new class of easily invertible circulant matrices, defined by exploiting the isomorphism from the ring Mnof n times n circulant matrices over GF(p) to the ring Rn= GF(p)[x]/(xn- 1) of the polynomials modulo (xn- 1). Such class contains matrices free of 4-length cycles that, if sparse, can be included in the parity check matrix of QC-LDPC codes. Bounds for the weight of their inverses are also determined, that are useful for designing sparse generator matrices for these error correcting codes.
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.
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 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.
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.
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.