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International Symposium on Turbo Codes and Iterative Information Processing

Codes that have tanner graphs with non-overlapping cycles

S. Srimathy, Andrew Thangaraj

Published Sep 1, 2008• 2 citations
Publisher

Abstract

The Sum-Product Algorithm (SPA) for the decoding of Low Density Parity Check (LDPC) codes produces exact posterior probabilities when the underlying Tanner graph is cycle-free. However, it has been shown that cycle-free Tanner graphs cannot support good codes as they have poor minimum distance properties. When a Tanner graph has cycles that do not overlap (no two cycles have any node in common), the corresponding code has managable minimal tree complexity so that optimal decoding can be achieved. In this paper, we consider codes whose Tanner graphs contain non-overlapping cycles. We derive upper bounds on the minimum distance for such codes, and show that the upper bound can be at most one more than that of a tree code for any given block length and rate. Our results imply that Tanner graphs of good codes will contain several cycles that overlap.

DOI
10.1109/turbocoding.2008.4658715

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The Sum-Product Algorithm (SPA) for the decoding of Low Density Parity Check (LDPC) codes produces exact posterior probabilities when the underlying Tanner graph is cycle-free. However, it has been shown that cycle-free Tanner graphs cannot support good codes as they have poor minimum distance properties. When a Tanner graph has cycles that do not overlap (no two cycles have any node in common), the corresponding code has managable minimal tree complexity so that optimal decoding can be achieved. In this paper, we consider codes whose Tanner graphs contain non-overlapping cycles. We derive upper bounds on the minimum distance for such codes, and show that the upper bound can be at most one more than that of a tree code for any given block length and rate. Our results imply that Tanner graphs of good codes will contain several cycles that overlap.
Short cycles have an important influence on iterative decoding for low-density parity-check(LDPC) codes.By using tree expanding of Tanner graphs and the analysis of the influence of invalid cycles and duplicate cycles,a formula for counting short cycles of LDPC codes is derived.With the derived formula,an algorithm for counting short cycles of Tanner graphs based on tree expanding is proposed.The proposed algorithm can count short cycles with arbitrary lengths,and thus overcomes the disadvantage of existing algorithms that count short cycles with given lengths.Numerical results of counting short cycles of some typical LDPC codes show the correctness of the proposed algorithm.
A code-independent performance bound for a given absorbing set is derived for quantized low-density parity-check (LDPC) decoders. The analysis demonstrates that each absorbing set in the Tanner graph imposes a specific lower bound on the frame error rate (FER) of any code containing that absorbing set under a given quantization scheme. This approach is applicable to any message-passing (MP) decoding algorithm and any uniform or non-uniform quantization scheme for LDPC codes. Simulation results using the sum-product algorithm (SPA) provide FERs that are consistent with the obtained bounds. In addition, the bounds demonstrate that the conventional quantized SPA is not capable of achieving very low FERs if the LDPC codes contain certain absorbing sets.
LDPC(Low Density Parity Check) codes are described by bipartite graphs with bit nodes and parity-check nodes. Tanner derived minimum distance bounds of the regular LDPC code in terms of the eigenvalues of the associated adjacency matrix. In this paper we generalize the Tanner's results. We derive minimum distance bounds applicable to both regular and blockwise-irregular LDPC codes. The first bound considers the relation between bit nodes in a minimum-weight codeword, and the second one considers the connectivity between parity nodes adjacent to a minimum-weight codeword. The derived bounds make it possible to describe the distance property of the code in terms of the eigenvalues of the associated matrix.
Quasi-cyclic (QC) low-density parity-check (LDPC) codes have several appealing properties regarding decoding, storage requirements and encoding aspects. In this paper, we focus on the QC LDPC codes over GF(q) whose parity-check matrices have fixed column weight j=2. By investigating two subgraphs in the Tanner graphs of the corresponding base matrices, we derive two upper bounds on the minimum Hamming distance for this class of codes. In addition, a method is proposed to construct QC LDPC codes over GF(q), which have good Hamming distance distributions. Simulations show that our designed codes have good performance.
Iterative decoding and linear programming decoding are guaranteed to converge to the maximum-likelihood codeword when the underlying Tanner graph is cycle-free. Therefore, cycles are usually seen as the culprit of low-density parity-check (LDPC) codes. In this paper, we argue in the context of graph cover pseudocodeword that, for a code that permits a cycle-free Tanner graph, cycles have no effect on error performance as long as they are a part of redundant rows. Specifically, we characterize all parity-check matrices that are pseudocodeword-free for such class of codes.
The linear block code is called a binary Low-Density Parity-Check code if whose parity-check matrix is a sparse matrix. For the features of parity-check matrix, a LDPC code can be illustrated by a Tanner graph. Cycles, especially short cycles in the Tanner graph, lead to inefficient decoding and prevent the decoding algorithm from converging to the optimal decoding result. In this paper, a simple algorithm of constructing four-cycle-free regular A(n,d_v,d_c)LDPC codes, d_v3, is introduced. And we analyse the relationship between the performance and the cycle.