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IEEE International Symposium on Information Theory (ISIT)

Tight Bounds of Minimum Distance Distributions of Irregular LDPC Code Ensembles

Shinya Miyamoto, Kenta Kasai, Tomoharu Shibuya, Kohichi Sakaniwa

Published Jun 1, 2007• 0 citations
Publisher

Abstract

Upper bounds of minimum distance distributions of Gallger codes and irregular LDPC codes were derived by Callage and Di, respectively. Di's bounds are tight for irregular LDPC codes which have variable nodes of degree two, however, it is not tight for irregular LDPC codes which do not. In this paper, we derive tight lower and upper bounds of minimum distance distributions of irregular LDPC code ensembles without variable nodes of degree two.

DOI
10.1109/isit.2007.4557315

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Upper bounds of minimum distance distributions of Gallger codes and irregular LDPC codes were derived by Callage and Di, respectively. Di's bounds are tight for irregular LDPC codes which have variable nodes of degree two, however, it is not tight for irregular LDPC codes which do not. In this paper, we derive tight lower and upper bounds of minimum distance distributions of irregular LDPC code ensembles without variable nodes of degree two.
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The asymptotic distance distribution of regular low-density parity-check (LDPC) codes derived by Litsin and Sherlev (2002) is generalised for the case of irregular LDPC codes with constant check node degree. Numerical analysis of simple irregular LDPC codes with weight-two variable nodes shows that their minimum Hamming distance can be increased linearly with the code length when the maximum variable node degree of the code is kept below a critical value. This result is consistent with the stability condition of the LDPC codes, at least for the simple irregular case.
This paper investigates about the usefulness of a recently published low complexity upper bound on beliefpropagation decoding thresholds for a class of irregular lowdensity parity-check (LDPC) codes. In particular, the class considered is characterized by variable node degree distributions λ(x) of minimum degree i1{i_1}i1​ >2: being, in this case, λ′\lambda^{\prime}λ′(0)=λ2=0\lambda_{2}=0λ2​=0, this is useful to design LDPC codes presenting a linear minimum distance growth with the block length with probability 1, as shown in Di et al.’s 2006 paper. These codes unfortunately cannot reach capacity under iterative decoding, since the achievement of capacity requires λ2≠0\lambda_{2}\neq0λ2​=0. However, in this latter case, the block error probability might converge to a constant, as shown in the aforementioned paper.
In this work, we present an upper bound on the minimum distance of array low-density parity-check (LDPC) codes. An array LDPC code is a quasi-cyclic LDPC code specified by two integers q and m, where q is an odd prime and m ≤ q. In the literature, the minimum distance of these codes (denoted by d(q,m)) has been thoroughly studied for m ≤ 5. Both exact results, for small values of q and m, and general (i.e., independent of q) bounds have been established. For m ≤ 6, the best known minimum distance upper bound, derived by Mittelholzer (IEEE Int. Symp. Inf. Theory, Jun./Jul. 2002), is d(q, 6) ≤ 32. In this work, we derive an improved upper bound of d(q, 6) ≤ 20 by using the concept of a template support matrix of a codeword. The bound is tight with high probability in the sense that we have not been able to find codewords of strictly lower weight for several values of q using a minimum distance probabilistic algorithm. Finally, we provide new specific minimum distance results for m ≤ 6 and low-to-moderate values of q ≤ 79.
Recently, LDPC codes with projected graph, or protograph structures have been proposed. In this paper, finite length ensemble weight enumerators for LDPC codes with protograph structures are obtained. Asymptotic results are derived as the block size goes to infinity. In particular, we are interested in obtaining ensemble average weight enumerators for protograph LDPC codes which have typical minimum distance that grows linearly with block size. As with irregular ensembles, linear minimum distance property is sensitive to the proportion of degree-2 variable nodes. In this paper, the derived results on ensemble weight enumerators show that linear minimum distance condition on degree distribution of unstructured irregular LDPC codes is a sufficient but not a necessary condition for protograph LDPC codes
In this letter, we obtain lower and upper bounds on the minimum distance dminof low-density parity-check (LDPC) codes. The bounds are derived by categorizing the non-zero code words of an LDPC code into two categories of elementary and non-elementary. The first category contains code words whose induced subgraph has only degree-2 check nodes. We propose an efficient search algorithm that can find the elementary code words of an LDPC code with weight less than a certain value amax, exhaustively. We also derive a lower bound Lneon the weight of non-elementary code words. By performing the search with amax= Lne, we either obtain an elementary code word with the smallest weight dmin, or establish the lower bound of Lneon dmin. For the upper bound, we modify our search algorithm to reach elementary codewords of larger weights at the cost of being non-exhaustive. Once such a codeword is found, its weight acts as an upper bound on dmin. We examine a large number of regular and irregular LDPC codes, and demonstrate the efficiency and versatility of our technique in finding lower and upper bounds on, and in many cases the exact value of, dmin. Finding dmin, or establishing search-based lower or upper bounds, for many of the examined codes are out of the reach of any existing algorithm.