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IEEE International Symposium on Information Theory (ISIT)

Which Codes Have 4-Cycle-Free Tanner Graphs?

Thomas R. Halford, K.M. Chugg, Alex J. Grant

Published Jul 1, 2006• 8 citations
Publisher

Abstract

Let C be an [n, k, d] binary linear code with rate R = k/n and dual Cperp. In this correspondence, it is shown that C can be represented by a 4-cycle-free Tanner graph only if: pdperples lfloorradicnp(p-1)+n2/4+n/2rfloor where p = n - k and dperpis the minimum distance of Cperp. By applying this result, it is shown that 4-cycle-free Tanner graphs do not exist for many classical binary linear block codes

DOI
10.1109/isit.2006.261717

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Let C be an [n, k, d] binary linear code with rate R = k/n and dual Cperp. In this correspondence, it is shown that C can be represented by a 4-cycle-free Tanner graph only if: pdperples lfloorradicnp(p-1)+n2/4+n/2rfloor where p = n - k and dperpis the minimum distance of Cperp. By applying this result, it is shown that 4-cycle-free Tanner graphs do not exist for many classical binary linear block codes
If a linear block code C of length n has a Tanner graph without cycles, then maximum-likelihood soft-decision decoding of C can be achieved in time O(n/sup 2/). However, we show that cycle-free Tanner graphs cannot support good codes. Specifically, let C be an (n,k,d) linear code of rate R=k/n that can be represented by a Tanner graph without cycles. We prove that if R/spl ges/0.5 then d/spl les/2, while if R<0.5 then C is obtained from a code of rate /spl ges/0.5 and distance /spl les/2 by simply repeating certain symbols. In the latter case, we prove that d/spl les/[n/k+1]+[n+1/k+1]<2/R. Furthermore, we show by means of an explicit construction that this bound is tight for all values of n and k. We also prove that binary codes which have cycle-free Tanner graphs belong to the class of graph-theoretic codes, known as cut-set codes of a graph. Finally, we discuss the asymptotics for Tanner graphs with cycles, and present a number of open problems for future research.
If a linear (n,k,d) code C has a Tanner graph without cycles, then maximum-likelihood soft-decision decoding of C can be achieved in time O(n/sup 2/). However, we show that cycle-free Tanner graphs cannot support good codes: we prove that d/spl les/max{2,2/R} for any linear code of rate R that can be represented by a cycle-free Tanner graph. Moreover, we explicitly construct codes that meet this bound with equality. We also show that all binary codes that have cycle-free Tanner graphs belong to the class of graph-theoretic cut-set codes. We conclude that the number of cycles in a Tanner graph must increase exponentially with the length n for asymptotically good codes.
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.
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.
Codes defined on graphs and their properties have been subjects of intense recent research. In this work, we are concerned with codes that have planar Tanner graphs. When the Tanner graph is planar, message-passing decoders can be efficiently implemented on chips without any issues of wiring. Also, recent work has shown the existence of optimal decoders for certain planar graphical models. The main contribution of this paper is an explicit upper bound on minimum distance ddd of codes that have planar Tanner graphs as a function of the code rate RRR for R≥5/8R \geq 5/8R≥5/8. The bound is given by\begin{equation*}d\le \left\lceil \frac{7-8R}{2(2R-1)} \right\rceil + 3\le 7.\end{equation*}As a result, high-rate codes with planar Tanner graphs will result in poor block-error rate performance, because of the constant upper bound on minimum distance.