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IEEE International Symposium on Information Theory (ISIT)

Minimum basis search algorithm for nearly maximum likelihood decoding of block codes

M. Ben-Yishai, Jakov Snyders

Published Dec 23, 2004• 0 citations
Publisher

Abstract

With any parity check matrix H of a code C(n,k,d) one may associate a graph whose vertices represent distinct bases, i.e., sets of n-k independent columns of H. Such basis specifies a unique error pattern. A search is carried out over a limited part of the bases graph for an error pattern having the minimum confidence value. Properties of the bases graph are utilized to reduce the complexity of the search procedure. Bounds are derived for the worst case computational complexity as well as for the correct-decoding probability.

DOI
10.1109/isit.2004.1365559

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With any parity check matrix H of a code C(n,k,d) one may associate a graph whose vertices represent distinct bases, i.e., sets of n-k independent columns of H. Such basis specifies a unique error pattern. A search is carried out over a limited part of the bases graph for an error pattern having the minimum confidence value. Properties of the bases graph are utilized to reduce the complexity of the search procedure. Bounds are derived for the worst case computational complexity as well as for the correct-decoding probability.
Summary Consider an (t1.k.d) binary linear block code specified by a parity-check matrix H. Assume that codewords with equal proba- bility are transmitted through a niemoryless channel. Let Y be the bit-by-bit hard-detected version of the received word, and assume that z, the corresponding syndrome, is nonzero. A set of 1 linearly independent columns of H that adds up to z will be called a /-pat- tern. Thus t does not exceed ni where n1 = ti-k is the number of check bits. A palier'ri is a I-pattern with unspecified 1. Assume, without essential loss of generality, that d t 3. Let r be the set of columns of H; then IF( = ri due to the aforementioned assumption. The ivei,oh/ of a column h of H is defined to be the confidence value (magnitude of the log likelihood ratio) of the bit associated with h. The weigh/ of a subset @ of r is the sum of the weights of the elements of @. An algorithm (I) for carrying out maximum likelihood decoding may now be phrased as follows: I) among all the /-patterns, with 1 ranging up to ni, find (the usually unique) one with least weight, then
A low-density parity-check code is a code specified by a parity-check matrix with the following properties: each column contains a small fixed numberj \geq 3of l's and each row contains a small fixed numberk > jof l's. The typical minimum distance of these codes increases linearly with block length for a fixed rate and fixedj. When used with maximum likelihood decoding on a sufficiently quiet binary-input symmetric channel, the typical probability of decoding error decreases exponentially with block length for a fixed rate and fixedj. A simple but nonoptimum decoding scheme operating directly from the channel a posteriori probabilities is described. Both the equipment complexity and the data-handling capacity in bits per second of this decoder increase approximately linearly with block length. Forj > 3and a sufficiently low rate, the probability of error using this decoder on a binary symmetric channel is shown to decrease at least exponentially with a root of the block length. Some experimental results show that the actual probability of decoding error is much smaller than this theoretical bound.
We introduce a new method for decoding short and moderate-length linear block codes with dense parity check matrix representations of cyclic form. This approach is termed multiple-bases belief-propagation. The proposed iterative scheme makes use of the fact that a code has many structurally diverse parity-check matrices, capable of detecting different error patterns. We show that this inherent code property leads to decoding algorithms with significantly better performance when compared to standard belief-propagation decoding. Furthermore, we describe how to choose sets of parity-check matrices of cyclic form amenable for multiple-bases decoding, based on analytical studies performed for the binary erasure channel. For several cyclic and extended cyclic codes, the multiple-bases belief propagation decoding performance can be shown to closely follow that of the maximum-likelihood decoder.
The minimum pseudocodeword weightwminof 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 distancedminunder 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 ofpseudoweightredundancyfor 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 havedmin=wmin. 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.
We present lower bounds on the bit error probability of binary linear block codes, and on the gap between the channel capacity and the rate of these codes for which reliable communica- tion is achievable. The results are valid for memo- ryless binary-input output-symmetric (MBIOS) chan- nels. The lower bounds provide a quantitative mea- sure for the number of cycles of bipartite graphs which represent good error-correcting codes. The tightness of the lower bounds is especially pronounced for the binary erasure channel (BEC); to this end, we analyze a sequence of ensembles of low-density parity-check (LDPC) codes which closely approach these bounds. I. Introduction If a linear block code can be represented by a cycle-free fac- tor graph which only includes variable nodes and parity-check nodes, then maximum-likelihood soft decision decoding can be performed with low complexity (1). Theorem 5 in (1) indicates that these graphs represent codes with very poor minimum distance. The bounds in (1) only refer to the minimum dis- tance of cycle-free codes, and in this work we derive lower bounds on the bit error probability and the gap to capacity for codes on graphs which are represented by bipartite graphs (with or without cycles). We derive in this work information- theoretic lower bounds on the bit error probability of a binary linear code used over MBIOS channels. The bounds are ex- pressed in terms of the density of an arbitrary parity-check matrix which represents the binary linear code (i.e., the num- ber of ones in the matrix normalized per information bit), and they are valid for codes whose bipartite graphs are with or without cycles. We introduce a quantitative measure for the cycles in a bipartite graph which represents a binary lin- ear code (where the graph only includes variable nodes and parity-check nodes). The bounds provide an interpretation for the tradeofi which exists between the bit error proba- bility and gap to capacity of an arbitrary LDPC code (i.e., its performance limitations), and the density of an arbitrary parity-check matrix which represents the code (where the lat- ter afiects the decoding complexity per information bit and per iteration, under a message-passing iterative decoding al- gorithm). We present in this work quantitative results which indicate that in order to approach the channel capacity with vanishing bit error probability, LDPC codes should not have too sparse parity-check matrices, as otherwise their inherent gap to capacity becomes large. In fact, we show that the density of any parity-check matrix and the fundamental num- ber of cycles of the graph which represents the code should increase at least like log 1 where designates the gap to ca- pacity under iterative message-passing decoding (see (2), (3)). The tightness of the lower bounds which are derived in this work (and whose validity is for any sequence of codes with 1 1.5 2 2.5 3 3.5 4 4.5