The structure of certain subgraphs of the Tanner graph of an LDPC code, the trapping sets, has been identified as important for the error floor performance of iterative decoding algorithms. To investigate such sets requires the parity check matrix of the code to be generated with sufficient structure that allows useful information to be obtained while giving good codes. Structures that have been considered include combinatorial designs and classical finite geometries. More recently other finite geometric notions such as partial geometries and generalized d-gons have been considered with some success. This work considers aspects of this approach.
The structure of certain subgraphs of the Tanner graph of an LDPC code, the trapping sets, has been identified as important for the error floor performance of iterative decoding algorithms. To investigate such sets requires the parity check matrix of the code to be generated with sufficient structure that allows useful information to be obtained while giving good codes. Structures that have been considered include combinatorial designs and classical finite geometries. More recently other finite geometric notions such as partial geometries and generalized d-gons have been considered with some success. This work considers aspects of this approach.
Several combinatorial properties of low-density parity-check (LDPC) codes, such as minimum distance, diameter, stopping number, girth and cycle-length distribution of the corresponding Tanner graph, are known to influence their performance under iterative decoding. Recently, a new class of combinatorial configurations, termed trapping sets, was shown to be of significant importance in determining the properties of LDPC codes in the error-floor region. Very little is known both about the existence/parameters of trapping sets in structured LDPC codes and about possible techniques for reducing their negative influence on the code's performance. In this paper, we address both these problems from an algorithmic and combinatorial perspective. We first provide a numerical study of the trapping phenomena for the Margulis code, which exhibits a fairly high error-floor. Based on this analysis, conducted for two different implementations of iterative belief propagation, we propose a novel decoding process, termed averaged decoding. Averaged decoding provides for a significant reduction in the number of incorrectly decoded frames in the error-floor region of the Margulis code. Furthermore, based on the results of the algorithmic approach, we suggest a novel combinatorial characterizations of trapping sets in the class of LDPC codes based on finite geometries. Projective geometry LDPC codes are suspected to have extremely low error-floors, which is a property that we may attribute to the non-existence of certain small trapping sets in the code graph.
In contrast to the capacity approaching performance of iteratively decoded low-density parity check (LDPC) codes, many practical finite-length LDPC codes exhibit performance degradation, manifested in a so-called error floor. Previous work has linked this phenomenon to the presence of certain combinatorial structures within the Tanner graph representation of the code, termed absorbing sets. Absorbing sets are stable under the bit-flipping operations and have been shown to act as fixed points (¿absorbers¿) for a wider class of iterative decoding algorithms. Codes often possess absorbing sets whose size is smaller than the minimum distance: the smallest absorbing sets are deemed most detrimental culprits behind the error floor. This paper focuses on the elementary combinatorial bounds of the smallest (candidate) absorbing sets. For certain classes of practical codes we demonstrate the tightness of these bounds and show how can the structure of the code and the structure of the absorbing sets be utilized to increase the size of the smallest absorbing sets without compromising other code properties such as the node degrees and the girth. As such, this work provides a step towards a better code design by taking into account the combinatorial nature of fixed points of iterative decoding algorithms.
The use of partial geometries to construct parity-check matrices for binary low-density parity-check (LDPC) codes has resulted in the design of successful codes with a probability of error on the AWGN channel close to the Shannon capacity at bit error rate down to 10−15 . Such considerations have motivated this further investigation. A new and simple construction of a type of partial geometries with a quasi-cyclic (QC) structure is given and their properties are investigated. Two new classes of this type of partial geometries, one based on prime fields and the other based on cyclic subgroups of prime orders of finite fields, are constructed. QC-LDPC codes with good error performances are constructed based on these two new classes of partial geometries. The trapping sets of the partial geometry codes were previously considered 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 on the AWGN channel under message-passing decoding.
It is well known that certain combinatorial structures in the Tanner graph of a low-density parity-check (LDPC) code exhibit a strong influence on its performance under iterative decoding. These structures include cycles, stopping/trapping sets, and parameters such as the diameter of the code. In general, it is very hard to find a complete characterization of such configurations in an arbitrary code, and even harder to understand the intricate relationships that exist between these entities. It is, therefore, of interest to identify a simple setting in which all the described combinatorial structures can be enumerated and studied within a joint framework. One such setting is developed in this paper, for the purpose of analyzing the distribution of short cycles and the structure of stopping and trapping sets in Tanner graphs of LDPC codes based on idempotent and symmetric Latin squares. The parity-check matrices of LDPC codes based on Latin squares have a special form that allows for connecting combinatorial parameters of the codes with the number of certain subrectangles in the Latin squares. Subrectangles of interest can be easily identified, and in certain instances, completely enumerated. This study can be extended in several different directions, one of which is concerned with modifying the code design process in order to eliminate or reduce the number of configurations bearing a negative influence on the performance of the code. Another application of the results includes determining to which extent a configuration governs the behavior of the bit-error rate curve in the waterfall and error-floor regions
The error-floor performance of low-density parity-check (LDPC) codes under iterative decoding is governed by combinatorial configurations in the Tanner graph of the code termed trapping sets. Finding the smallest trapping set in a Tanner graph is an NP-hard problem. However, for codes constructed from designs one can partially characterize trapping sets and enumerate them efficiently. We focus on LDPC codes based on Steiner triple systems (STS), and quantify small trapping sets for bit-flipping decoding over the BSC and small trapping sets for the AWGN channel. Furthermore, we provide simulation results that show that the enumeration scheme at hand provides good estimates for the error-floor behavior of STS LDPC codes.
Error correcting codes are used in virtually all communication systems to ensure reliable transmission of information. In 1948, Shannon established an upper-bound on the maximum rate at which information can be transmitted reliably over a noisy channel. Reliably transmitting information with a rate close to this theoretical limit, known as the channel capacity, has been the goal of channel coding scientists for the last five decades. The rediscovery of low-density parity-check (LDPC) codes in the 1990s added much-renewed excitement in the coding community. LDPC codes are interesting because they can approach channel capacity under sub-optimum decoding algorithms whose complexity is linear in the code length. Unsurprisingly, LDPC codes quickly attained their popularity in practical applications such as magnetic storage, wireless and optical communications. One, if not the most, important and challenging problem in LDPC code research is the study and analysis of the error floor phenomenon. This phenomenon is described as an abrupt degradation in the frame error rate performance of LDPC codes in the high signal-to-noise ratio region. Error floor is harmful because its presence prevents the LDPC decoder from reaching very low probability of decoding failure, an important requirement for many applications. Not long after the rediscovery of LDPC codes, scientists established that error floor is caused by certain harmful structures, most commonly known as trapping sets, in the Tanner representation of a code. Since then, the study of error floor mostly consists of three major problems: 1) estimating error floor; 2) constructing LDPC codes with low error floor and 3) designing decoders that are less susceptible to error floor. Although some parts of this dissertation can be used as important elements in error floor estimation, our main contributions are a novel method for constructing LDPC codes with low error floor and a novel class of low complexity decoding algorithms that can collectively alleviate error floor. These contributions are summarized as follows. A method to construct LDPC codes with low error floors on the binary symmetric channel is presented. Codes are constructed so that their Tanner graphs are free of certain small trapping sets. These trapping sets are selected from the Trapping Set Ontology for the Gallager A/B decoder. They are selected based on their relative harmfulness for a given decoding algorithm. We evaluate the relative harmfulness of different trapping sets for the sum-product algorithm by using the topological relations among them and by analyzing the decoding failures on one trapping set in the presence or absence of other trapping sets. We apply this method to construct structured LDPC codes. To facilitate the discussion, we give a new description of structured LDPC codes whose parity-check matrices are arrays of permutation matrices. This description uses Latin squares to define a set of permutation matrices that have disjoint support and to derive a simple necessary and sufficient condition for the Tanner graph of a code to be free of four-cycles. A new class of bit flipping algorithms for LDPC codes over the binary symmetric channel is proposed. Compared to the regular (parallel or serial) bit flipping algorithms, the proposed algorithms employ one additional bit at a variable node to represent its "strength." The introduction of this additional bit allows an increase in the guaranteed error correction capability. An additional bit is also employed at a check node to capture information which is beneficial to decoding. A framework for failure analysis and selection of two-bit bit flipping algorithms is provided. The main component of this framework is the (re)definition of trapping sets, which are the most "compact" Tanner graphs that cause decoding failures of an algorithm. A recursive procedure to enumerate trapping sets is described. This procedure is the basis for selecting a collection of algorithms that work well together. It is demonstrated that decoders which employ a properly selected group of the proposed algorithms operating in parallel can offer high speed and low error floor decoding.