An Improved Method for Counting 6-Cycles in Low-Density Parity-Check Codes

Abstract

Since their rediscovery in the early 1990s, low-density parity-check (LDPC) codes have become the most popular error-correcting codes owing to their excellent performance. An LDPC code is a linear block code that has a sparse parity-check matrix. Cycles in this matrix, particularly short cycles, degrade the performance of such a code. Hence, several methods for counting short cycles in LDPC codes have been proposed, such as Fan’s method to detect 4-cycles, 6- cycles, 8-cycles, and 10-cycles. Unfortunately, this method fails to count all 6- cycles, i.e., ignores numerous 6-cycles, in some given parity-check matrices. In this paper, an improvement of this algorithm is presented that detects all 6-cycles in LDPC codes, as well as in general bipartite graphs. Simulations confirm that the improved method offers the exact number of 6-cycles, and it succeeds in detecting those ignored by Fan’s method.