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IEEE Transactions on Information Theory

Tree-Based Construction of LDPC Codes Having Good Pseudocodeword Weights

Christine A. Kelley, Deepak Sridhara, Joachim Rosenthal

Published Apr 1, 2007• 18 citations
Publisher

Abstract

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$d=p^s$and$d = p^s+1$, 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$p > 2$.

DOI
10.1109/tit.2007.892774

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IEEE Transactions on Information Theory•Oct 24, 2005•41 citations
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 addd-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 degreesd=psd=p^sd=psandd=ps+1d = p^s+1d=ps+1, forpppa 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 forppp-ary LDPC codes. Treating these codes asppp-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 wherep>2p > 2p>2.
We present a tree-based construction of 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 ddd-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=psd=p^sd=ps and d=ps+1d = p^s+1d=ps+1, for ppp 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 ppp-ary LDPC codes. Treating these codes as ppp-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>2p > 2p>2.
An independent tree-based method for lower bounding the minimum distance of low-density parity-check (LDPC) codes is presented. This lower-bound is then used as the decision criterion during the iterative construction of regular LDPC codes. The new construction algorithm results in LDPC codes with greater girth and improved minimum-distance bounds when compared to regular LDPC codes constructed using the progressive edge-growth (PEG) construction and the approximate cycle extrinsic message degree (ACE)-constrained PEG construction. Simulation results of codes constructed with the new method show improved performance on the additive white Gaussian noise channel at moderate signal-to-noise ratios.
This paper presents a geometric approach to the construction of low-density parity-check (LDPC) codes. Four classes of LDPC codes are constructed based on the lines and points of Euclidean and projective geometries over finite fields. Codes of these four classes have good minimum distances and their Tanner (1981) graphs have girth 6. Finite-geometry LDPC codes can be decoded in various ways, ranging from low to high decoding complexity and from reasonably good to very good performance. They perform very well with iterative decoding. Furthermore, they can be put in either cyclic or quasi-cyclic form. Consequently, their encoding can be achieved in linear time and implemented with simple feedback shift registers. This advantage is not shared by other LDPC codes in general and is important in practice. Finite-geometry LDPC codes can be extended and shortened in various ways to obtain other good LDPC codes. Several techniques of extension and shortening are presented. Long extended finite-geometry LDPC codes have been constructed and they achieve a performance only a few tenths of a decibel away from the Shannon theoretical limit with iterative decoding.
In 1948, Claude Shannon provided a sound mathematical foundation for communication, and proved the existence of good codes. Low-density parity-check (LDPC) codes are a family of error-correcting codes that achieve reliable performance (i.e., arbitrarily low probability of decoding error) near the Shannon-limit when decoded iteratively via very efficient graph-based message-passing decoders, the complexity of which grows only linearly in the block-length. Despite their tremendous success, LDPC codes lack a strong theoretical foundation and few explicit constructions of them are known that outperform their random counterparts. The first part of this thesis investigates how the structure of a graph representing an LDPC code affects the decoding performance. In particular, lower bounds on the minimum pseudocodeword weight, a key parameter in predicting iterative decoding performance, are obtained which link the performance of the code to the structural properties of its graph. One bound is a tree bound and is extended to generalized and p-ary LDPC codes. To this end, a suitable definition of pseudocodeword weight is derived for the p-ary symmetric channel. Moreover, the relationship between pseudocodeword weight distributions and different graph representations is examined. Bounds on the minimum pseudocodeword weight are also presented for several classes of expander codes based on the eigenvalues of the adjacency matrix of the Tanner graph. The second part of this thesis constructs LDPC codes having desirable properties. The first construction is combinatorial and uses permutations and mutually orthogonal Latin squares. One type results in the well-known projective-geometrybased LDPC codes. Another type yields a new family of codes having comparable parameters and pseudocodeword weights suitable for iterative decoding. The resulting performance of these codes is superior in comparison with that of comparable random LDPC codes, a feature attributed to the lack of low-weight bad pseudocodewords. For the second construction, unbalanced expander graphs are obtained by modifying the original zig-zag graph product to take bi-regular expander graphs as components, with suitable constraints on the degrees. Graphs resulting from the zig-zag and replacement product graphs are then used in the design of generalized LDPC codes having compact representation and pseudocodeword weights lower-bounded by the eigenvalue bounds.
We use the theory of finite classical generalized polygons to derive and study low-density parity-check (LDPC) codes. The Tanner graph of a generalized polygon LDPC code is highly symmetric, inherits the diameter size of the parent generalized polygon, and has minimum (one half) diameter-to-girth ratio. We show formally that when the diameter is four or six or eight, all codewords have even Hamming weight. When the generalized polygon has in addition an equal number of points and lines, we see that the nonregular polygon based code construction has minimum distance that is higher at least by two in comparison with the dual regular polygon code of the same rate and length. A new minimum-distance bound is presented for codes from nonregular polygons of even diameter and equal number of points and lines. Finally, we prove that all codes derived from finite classical generalized quadrangles are quasi-cyclic and we give the explicit size of the circulant blocks in the parity-check matrix. Our simulation studies of several generalized polygon LDPC codes demonstrate powerful bit-error-rate (BER) performance when decoding is carried out via low-complexity variants of belief propagation.