Deepak Sridhara

LDPC block and convolutional codes based on circulant matrices

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.

Pseudocodewords of Tanner Graphs

This paper presents a detailed analysis of pseudocodewords of Tanner graphs. Pseudocodewords arising on the iterative decoders computation tree are distinguished from pseudocodewords arising on finite degree lifts. Lower bounds on the minimum pseudocodeword weight are presented for the BEC, BSC, and AWGN channel. Some structural properties of pseudocodewords are examined, and pseudocodewords and graph properties that are potentially problematic with min-sum iterative decoding are identified. An upper bound on the minimum degree lift needed to realize a particular irreducible lift-realizable pseudocodeword is given in terms of its maximal component, and it is shown that all irreducible lift-realizable pseudocodewords have components upper bounded by a finite value t that is dependent on the graph structure. Examples and different Tanner graph representations of individual codes are examined and the resulting pseudocodeword distributions and iterative decoding performances are analyzed. The results obtained provide some insights in relating the structure of the Tanner graph to the pseudocodeword distribution and suggest ways of designing Tanner graphs with good minimum pseudocodeword weight.

LDPC codes over rings for PSK modulation

This paper describes the design and analysis of low-density parity-check (LDPC) codes over rings and shows how these codes, when mapped onto appropriate signal constellations, can be used to effect bandwidth-efficient modulation. Specifically, LDPC codes are constructed over the integer rings /spl Zopf//sub m/ and G/sub m//sup 2/ and mapped onto phase-shift keying (PSK)-type signal sets to yield geometrically uniform signal space codes. This paper identifies and addresses the design issues that affect code performance. Examples of codes over /spl Zopf//sub 8/ and G/sub 64/ mapped onto 8-ary and 64-ary signal sets at a spectral efficiency of 1.5 and 2.0 bits per second per hertz (b/s/Hz) illustrate the approach; simulation of these codes over the additive white Gaussian noise (AWGN) channel demonstrates that this approach is a good alternative to bandwidth-efficient techniques based on binary LDPC codes-e.g., bit-interleaved coded modulation.

Pseudocodeword weights and stopping sets

We examine the structure of pseudocodewords in Tanner graphs and derive lower bounds of pseudocodeword weights. The weight of a pseudocodeword is related to the size of its support set, which forms a stopping set in the Tanner graph.

Tree-Based Construction of LDPC Codes Having Good Pseudocodeword Weights

We present a tree-based construction of low-density parity-check (LDPC) codes that have minimum pseudocodeword weight equal to or almost equal to the minimum distance, and perform well with iterative decoding. The construction involves enumerating a d-regular tree for a fixed number of layers and employing a connection algorithm based on permutations or mutually orthogonal Latin squares to close the tree. Methods are presented for degrees d=ps and d=ps+1, for p a prime. One class corresponds to the well-known finite-geometry and finite generalized quadrangle LDPC codes; the other codes presented are new. We also present some bounds on pseudocodeword weight for p-ary LDPC codes. Treating these codes as p-ary LDPC codes rather than binary LDPC codes improves their rates, minimum distances, and pseudocodeword weights, thereby giving a new importance to the finite-geometry LDPC codes where p>2

Pseudocodeword weights for non-binary LDPC codes

Pseudocodewords of q-ary LDPC codes are examined and the weight of a pseudocodeword on the q-ary symmetric channel is defined. The weight definition of a pseudocodeword on the AWGN channel is also extended to two-dimensional q-ary modulation such as q-PAM and q-PSK. The tree-based lower bounds on the minimum pseudocodeword weight are shown to also hold for q-ary LDPC codes on these channels

Zig-zag and replacement product graphs and LDPC codes

It is known that the expansion property of a graph influences the performance of the corresponding code when decoded using iterative algorithms. Certain graph products may be used to obtain larger expander graphs from smaller ones. In particular, the zig-zag product and replacement product may be used to construct infinite families of constant degree expander graphs. This paper investigates the use of zig-zag and replacement product graphs for the construction of codes on graphs. A modification of the zig-zag product is also introduced, which can operate on two unbalanced biregular bipartite graphs, and a proof of the expansion property of this modified zig-zag product is presented.

On the pseudocodeword weight and parity-check matrix redundancy of linear codes

The minimum pseudocodeword weight w min of a linear graph-based code is more influential in determining decoding performance when decoded via iterative and linear programming decoding algorithms than the classical minimum distance d min under standard maximum-likelihood decoding Moreover, unlike the minimum distance which is unique to the code regardless of representation, the set of pseudocodewords, and therefore also the minimum pseudocodeword weight, depends on the graph representation used in decoding as well as on the communication channel. This means that a judicious choice of parity-check matrix is crucial for realizing the best potential of any graph-based code. In this paper, we introduce the notion of pseudoweight redundancy for the memoryless binary symmetric channel (BSC). Analogous to the stopping redundancy in the literature, this parameter gives the smallest number of rows needed for a parity-check matrix to have d min = w min. We provide some upper bounds on the BSC-pseudoweight redundancy and illustrate the concept with some results for Hamming codes, tree-based and finite geometry LDPC codes, Reed-Muller codes and Hadamard codes.

A construction for irregular low density parity check convolutional codes

A technique for constructing irregular low density parity check convolutional codes is described. The constructed codes exhibit lower convergence thresholds with belief propagation decoding than their regular counterparts.

EIGENVALUE BOUNDS ON THE PSEUDOCODEWORD WEIGHT OF EXPANDER CODES

Four different ways of obtaining low-density parity-check codes from expander graphs are considered. For each case, lower bounds on the minimum stopping set size and the minimum pseudocodeword weight of expander (LDPC) codes are derived. These bounds are compared with the known eigenvalue-based lower bounds on the minimum distance of expander codes. Furthermore, Tanners parity-oriented eigenvalue lower bound on the minimum distance is generalized to yield a new lower bound on the minimum pseudocodeword weight. These bounds are useful in predicting the performance of LDPC codes under graph-based iterative decoding and linear programming decoding.