Error Rate Estimation of Finite-Length Low-Density Parity-Check Codes on Binary Symmetric Channels Using Cycle Enumeration

Abstract

The performance of low-density parity-check (LDPC) codes decoded by hard-decision iterative decoding algorithms can be accurately estimated if the weight J and the number IE of the smallest error patterns that cannot be corrected by the decoder are known. To obtain J and IEj , one would need to perform the direct enumeration of error patterns with weight i < J. The complexity of enumeration increases exponentially with J, essentially as ni, where n is the code block length. In this paper, we approximate J and IEj by enumerating and testing the error patterns that are subsets of short cycles in the code's Tanner graph. This reduces the computational complexity by several orders of magnitude compared to direct enumeration, making it possible to estimate the error rates for almost any practical LDPC code. To obtain the error rate estimates, we propose an algorithm that progressively improves the estimates as larger cycles are enumerated. Through a number of examples, we demonstrate that the proposed method can accurately estimate both the bit error rate (BER) and the frame error rate (FER) of regular and irregular LDPC codes decoded by a variety of hard-decision iterative decoding algorithms.

Error rate estimation of finite-length low-density parity-check codes decoded by soft-decision iterative algorithms

“…In cycle enumeration [10], we enumerate cycles of different lengths starting from those in C g . For a q-bit decoder, for each cycle, input vectors are generated by setting the vector elements equal to 2 q−1 −1 in all positions except those involved in the cycle.…”

“…As error patterns with larger distances are examined the estimates improve. Similar to the approach of [10], one can in fact stop the algorithm when two successive estimates are close together indicating a convergent behavior.…”

“…This method provides very accurate estimates of both the FER and the bit error rate (BER) over the whole range of crossover probabilities of practical interest. To reduce the computational complexity of the direct enumeration used in [9], we proposed a cycle enumeration algorithm in [10] which reduced the computational complexity by up to several orders of magnitude.…”

Performance estimation, code construction and decoding of finite-length low-density parity-check codes

“…The results of this thesis have been partly published or submitted for publication in a number of conference proceedings and journals [18][19][20][21][22][23][24][25][26][27][28],…”

“…To obtain the error rate estimates, we propose an algorithm that progressively improves the estimates as larger cycles are enumerated. These results have been published in [21] and submitted for possible publication in IEEE Trans. Comm.…”