New Classes of Partial Geometries and Their Associated LDPC Codes

The use of partial geometries to construct parity-check matrices for LDPC codes has resulted in the design of successful codes with a probability of error close to the Shannon capacity at bit error rates down to $10^{-15}$. Such considerations have motivated this further investigation. A new and simple construction of a type of partial geometries with quasi-cyclic structure is given and their properties are investigated. The trapping sets of the partial geometry codes were considered previously using the geometric aspects of the underlying structure to derive information on the size of allowable trapping sets. This topic is further considered here. Finally, there is a natural relationship between partial geometries and strongly regular graphs. The eigenvalues of the adjacency matrices of such graphs are well known and it is of interest to determine if any of the Tanner graphs derived from the partial geometries are good expanders for certain parameter sets, since it can be argued that codes with good geometric and expansion properties might perform well under message-passing decoding.Comment: 34 pages with single column, 6 figure

LDPC codes from semipartial geometries

A binary low-density parity-check (LDPC) code is a linear block code that is defined by a sparse parity-check matrix H, that is H has a low density of 1’s. LDPC codes were originally presented by Gallager in his doctoral dissertation [9], but largely overlooked for the next 35 years. A notable exception was [29], in which Tanner introduced a graphical representation for LDPC codes, now known as Tanner graphs. However, interest in these codes has greatly increased since 1996 with the publication of [22] and other papers, since it has been realised that LDPC codes are capable of achieving near-optimal performance when decoded using iterative decoding algorithms. LDPC codes can be constructed randomly by using a computer algorithm to generate a suitable matrix H. However, it is also possible to construct LDPC codes explicitly using various incidence structures in discrete mathematics. For example, LDPC codes can be constructed based on the points and lines of finite geometries: there are many examples in the literature (see for example [18, 28]). These constructed codes can possess certain advantages over randomly-generated codes. For example they may provide more efficient encoding algorithms than randomly-generated codes. Furthermore it can be easier to understand and determine the properties of such codes because of the underlying structure. LDPC codes have been constructed based on incidence structures known as partial geometries [16]. The aim of this research is to provide examples of new codes constructed based on structures known as semipartial geometries (SPGs), which are generalisations of partial geometries. Since the commencement of this thesis [19] was published, which showed that codes could be constructed from semipartial geometries and provided some examples and basic results. By necessity this thesis contains a number of results from that paper. However, it should be noted that the scope of [19] is fairly limited and that the overlap between the current thesis and [19] is consequently small. [19] also contains a number of errors, some of which have been noted and corrected in this thesis

Low-Density Parity-Check Codes From Transversal Designs With Improved Stopping Set Distributions

This paper examines the construction of low-density parity-check (LDPC) codes from transversal designs based on sets of mutually orthogonal Latin squares (MOLS). By transferring the concept of configurations in combinatorial designs to the level of Latin squares, we thoroughly investigate the occurrence and avoidance of stopping sets for the arising codes. Stopping sets are known to determine the decoding performance over the binary erasure channel and should be avoided for small sizes. Based on large sets of simple-structured MOLS, we derive powerful constraints for the choice of suitable subsets, leading to improved stopping set distributions for the corresponding codes. We focus on LDPC codes with column weight 4, but the results are also applicable for the construction of codes with higher column weights. Finally, we show that a subclass of the presented codes has quasi-cyclic structure which allows low-complexity encoding.Comment: 11 pages; to appear in "IEEE Transactions on Communications

LDPC codes associated with linear representations of geometries

We look at low density parity check codes over a finite field K associated with finite geometries T*(2) (K), where K is any subset of PG(2, q), with q = p(h), p not equal char K. This includes the geometry LU(3, q)(D), the generalized quadrangle T*(2)(K) with K a hyperoval, the affine space AG(3, q) and several partial and semi-partial geometries. In some cases the dimension and/or the code words of minimum weight are known. We prove an expression for the dimension and the minimum weight of the code. We classify the code words of minimum weight. We show that the code is generated completely by its words of minimum weight. We end with some practical considerations on the choice of K

LDPC codes from the Hermitian curve

In this paper, we study the code C which has as parity check matrix H the incidence matrix of the design of the Hermitian curve and its (q + 1)-secants. This code is known to have good performance with an iterative decoding algorithm, as shown by Johnson and Weller in ( Proceedings at the ICEE Globe com conference, Sanfrancisco, CA, 2003). We shall prove that C has a double cyclic structure and that by shortening in a suitable way H it is possible to obtain new codes which have higher code-rate. We shall also present a simple way to constructing the matrix H via a geometric approach

Incidence structures from the blown-up plane and LDPC codes

In this article, new regular incidence structures are presented. They arise from sets of conics in the affine plane blown-up at its rational points. The LDPC codes given by these incidence matrices are studied. These sparse incidence matrices turn out to be redundant, which means that their number of rows exceeds their rank. Such a feature is absent from random LDPC codes and is in general interesting for the efficiency of iterative decoding. The performance of some codes under iterative decoding is tested. Some of them turn out to perform better than regular Gallager codes having similar rate and row weight.Comment: 31 pages, 10 figure

A Simplified Min-Sum Decoding Algorithm for Non-Binary LDPC Codes

Non-binary low-density parity-check codes are robust to various channel impairments. However, based on the existing decoding algorithms, the decoder implementations are expensive because of their excessive computational complexity and memory usage. Based on the combinatorial optimization, we present an approximation method for the check node processing. The simulation results demonstrate that our scheme has small performance loss over the additive white Gaussian noise channel and independent Rayleigh fading channel. Furthermore, the proposed reduced-complexity realization provides significant savings on hardware, so it yields a good performance-complexity tradeoff and can be efficiently implemented.Comment: Partially presented in ICNC 2012, International Conference on Computing, Networking and Communications. Accepted by IEEE Transactions on Communication

Absorbing Set Analysis and Design of LDPC Codes from Transversal Designs over the AWGN Channel

In this paper we construct low-density parity-check (LDPC) codes from transversal designs with low error-floors over the additive white Gaussian noise (AWGN) channel. The constructed codes are based on transversal designs that arise from sets of mutually orthogonal Latin squares (MOLS) with cyclic structure. For lowering the error-floors, our approach is twofold: First, we give an exhaustive classification of so-called absorbing sets that may occur in the factor graphs of the given codes. These purely combinatorial substructures are known to be the main cause of decoding errors in the error-floor region over the AWGN channel by decoding with the standard sum-product algorithm (SPA). Second, based on this classification, we exploit the specific structure of the presented codes to eliminate the most harmful absorbing sets and derive powerful constraints for the proper choice of code parameters in order to obtain codes with an optimized error-floor performance.Comment: 15 pages. arXiv admin note: text overlap with arXiv:1306.511