Performance of low density parity check codes as a function of actual and assumed noise levels

We investigate how sensitive Gallager’s codes are, when decoded by the sum-product algorithm, to the assumed noise level. We have found a remarkably simple function that fits the empirical results as a function of the actual noise level at both high and low noise levels. A linear code may be described in terms of a generator matrix G or in terms of a parity check matrix H, which satisfies Hx = 0 for all codewords x. In 1962, Gallager reported work on binary codes defined in terms of low density parity check matrices [2,3]. The matrix H is defined in a non-systematic form; each column of H has the same small predetermined weight (e.g., 3) and the weight per row is also uniform; the matrix H is constructed at random subject to these constraints. When these codes are decoded using Gallager’s iterative probabilistic decoding method, also known as the sum-product algorithm or belief propagation, their empirical performance is found to be excellent [4,5]. In this paper, we investigate the dependence of the performance of a low density parity check code on both the assumed and actual noise levels of a binary symmetric channel and a Gaussian channel. All the experiments make use of binary codes whose parity check matrices are randomly generated with weight per column t = 3 using construction methods 1A and 1B of [4]. The present work has four motivations. First, since the decoding algorithm (reviewed in detail in [4,5]) is somewhat ad hoc, it seems an interesting experiment to ‘tweak’ it, checking to see whether any of its control parameters can be set to better values than their standard values; specifically, we tweak the assumed noise level to see if setting it to higher or lower values improves the performance of the sum-product algorithm. Second, this project sheds 1 Email: [email protected] c ©2003 Published by Elsevier Science B. V.