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Information Theory and Applications Workshop (ITA)

On the existence of typical minimum distance for protograph-based LDPC codes

Shadi Abu‐Surra, D. Divsalar, W.E. Ryan

Published Jan 1, 2010• 33 citations
Publisher

Abstract

In this paper we prove that, for a certain class of protograph-based LDPC codes with degree-two variable nodes, a typical minimum distance exists. We obtain a tight bound on the sum of weight enumerators, up to some weight d*, for the ensemble of finite-length protograph LDPC codes. Then we prove that this sum goes to zero as the block length goes to infinity. Finally, we prove that Pr(d < d*) goes to zero as the block length goes to infinity. This typical minimum distance exists if degree-two nodes have certain connections to the check nodes. This is also important in practice since it identifies a certain class of protograph LDPC codes that have typical minimum distances.

DOI
10.1109/ita.2010.5454136

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In this paper we prove that, for a certain class of protograph-based LDPC codes with degree-two variable nodes, a typical minimum distance exists. We obtain a tight bound on the sum of weight enumerators, up to some weight d*, for the ensemble of finite-length protograph LDPC codes. Then we prove that this sum goes to zero as the block length goes to infinity. Finally, we prove that Pr(d < d*) goes to zero as the block length goes to infinity. This typical minimum distance exists if degree-two nodes have certain connections to the check nodes. This is also important in practice since it identifies a certain class of protograph LDPC codes that have typical minimum distances.
This paper proves the existence of the typical minimum distance for certain ensembles of the protograph-based nonbinary (PB NB) LDPC codes with degree-2 variable nodes. Using recently obtained ensemble weight enumerators for the PB NB LDPC codes, we show that asymptotically in the code length N, the entries in the weight enumerators up to some weight d* = δ* N (δ* >; 0) vanish under certain conditions on degree-2 nodes. As a consequence, the probability that the code minimum distance is less than d* goes to zero as the code length goes to infinity. Results of this type advance the understanding of structured non-binary LDPC codes.
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
Protograph-based generalized LDPC (GLDPC) codes have the advantages of a simple design procedure and highly structured encoders and decoders. Recently, a technique for computing ensemble weight enumerators for GLDPC codes has been published. In the current paper, we investigate the existence of typical minimum distance for protograph-based GLDPC codes. That is, we first upper bound the ensemble weight enumerators for finite-length GLDPC codes based on protographs, and then we consider the sum of weight enumerators. The results allow us to determine whether or not the typical minimum distance in the ensemble grows linearly with codeword length. We provide conditions on the connections of degree-2 variable nodes to constraint nodes (short block codes) to have typical minimum distance. These conditions are related to the minimum distances of the constraint nodes.
Protograph-based LDPC codes have the advantages of a simple design (or search) procedure and highly structured encoders and decoders. These advantages have also been exploited in the design of protograph-based generalized LDPC (G-LDPC) codes. Recently, a technique for computing ensemble weight enumerators for protograph-based LDPC codes has been published. In the current paper, we extend those results to protograph-based G-LDPC codes. That is, we first derive ensemble weight enumerators for finite-length G-LDPC codes based on protographs, and then we consider the asymptotic case. The asymptotic results allow us to determine whether or not the typical minimum distance in the ensemble grows linearly with codeword length.
Recently, pseudocodewords of Tanner graphs of LDPC codes have been used to explain the behavior of iterative decoders for these codes. In this paper, finite-length pseudocodeword weight enumerators for protograph-based generalized-LDPC code ensembles are obtained. Then asymptotic results are derived from the finite-length results by letting the block length go to infinity. The asymptotic results allow us to determine whether or not the typical minimum pseudocodeword weight grows linearly with codeword length. We give examples with Hamming component codes.
Protograph-based LDPC codes have the advantages of a simple design (or search) procedure and highly structured encoders and decoders. These advantages have also been exploited in the design of protograph-based generalized LDPC (G-LDPC) codes. Recently, a technique for computing ensemble weight enumerators and stopping set enumerators for protograph-based LDPC codes has been published. In the current paper, we extend those results to protograph-based G-LDPC codes. That is, we first derive ensemble weight and stopping set enumerators for finite-length G-LDPC codes based on protographs, and then we consider the asymptotic case. In the weight enumerator case, the asymptotic results allow us to determine whether or not the typical minimum distance in the ensemble grows linearly with codeword length. In the stopping set enumerator case, the asymptotic results allows us to determine whether or not the typical smallest stopping set size grows linearly with codeword length.