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arXiv:cs/0605118 · IEEE International Symposium on Information Theory (ISIT)

Pseudocodeword weights for non-binary LDPC codes

Christine Kelley, Deepak Sridhara, Joachim Rosenthal

Published Jul 1, 2006• 8 citations
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Abstract

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

DOI
10.1109/isit.2006.262072

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IEEE Transactions on Information Theory•Apr 1, 2007•18 citations
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
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.
In this correspondence, we study the minimum pseudoweight and minimum pseudocodewords of low-density parity-check (LDPC) codes under linear programming (LP) decoding. First, we show that the lower bound of Kelley, Sridhara, Xu, and Rosenthal on the pseudoweight of a nonzero pseudocodeword of an LDPC code whose Tanner graph has girth greater than is tight if and only if this pseudocodeword is a real multiple of a codeword. Then, the lower bound of Kashyap and Vardy on the stopping distance of an LDPC code is proved to be also a lower bound on the pseudoweight of a nonzero pseudocodeword of an LDPC code whose Tanner graph has girth , and this lower bound is tight if and only if this pseudocodeword is a real multiple of a codeword. Using these results we further obtain that for some LDPC codes, there are no other minimum pseudocodewords except the real multiples of minimum weight codewords. This means that the LP decoding for these LDPC codes is asymptotically optimal in the sense that the ratio of the probabilities of decoding errors of LP decoding and maximum-likelihood decoding approaches as the signal-to-noise ratio (SNR) tends to infinity. Finally, some LDPC codes are listed to illustrate these results.
The concepts of pseudocodeword and pseudoweight play a fundamental role in the finite-length analysis of LDPC codes. The pseudoredundancy of a binary linear code is defined as the minimum number of rows in a parity-check matrix such that the corresponding minimum pseudoweight equals its minimum Hamming distance. By using the value assignment of Chen and Kløve we present new results on the pseudocodeword redundancy of binary linear codes. In particular, we give several upper bounds on the pseudoredundancies of certain codes with repeated and added coordinates and of certain shortened subcodes. We also investigate several kinds of k-dimensional binary codes and compute their exact pseudocodeword redundancy.
Pseudocodewords, and in particular minimal pseudocodewords, play an important role in understanding the performance of linear programming (LP) decoding. In this paper, we investigate minimal pseudocodewords of binary Hamming codes described by full-rank parity-check matrices. We first provide some general results on minimal pseudocodewords with support size 3 of a binary parity-check matrix. We also prove a lower bound on the minimum binary symmetric channel (BSC) pseudoweight of a binary parity-check matrix. Then we prove that a full-rank parity-check matrix of a binary Hamming code has minimal pseudocodewords of certain types whose support sizes are larger than 3. Interestingly enough, the BSC pseudoweight of all these minimal pseudocodewords is 2. Using this fact as well as the above-mentioned lower bound, we further prove that a full-rank parity-check matrix of a binary Hamming code has minimum BSC pseudoweight 2. Moreover, the additive white Gaussian noise channel (AWGNC) pseudoweight of all these minimal pseudocodewords is 3. Based on numerical observations, we conjecture that a full-rank parity-check matrix of a binary Hamming code has minimum AWGNC pseudoweight 3. Finally, we provide more properties of a subset of minimal pseudocodewords of a full-rank parity-check matrix of a binary Hamming code.
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.