Error exponents for recursive decoding of Reed-Muller codes

Abstract

Recursive decoding is studied for Reed-Muller (RM) codes used on a binary symmetric channel. Decoding is performed beyond the bounded distance radius d/2 and corrects most error patterns of weight up to (d ln d)/2. In our analysis, coding is decomposed into consecutive steps, with one information bit derived in each step. Then the error probability of each step is defined by the recursive recalculations of the Bernoulli random variables. We derive the exponential moments of the recalculated random variables. As a result, tight exponential bounds on the output error probability are obtained for the two recursive algorithms considered in the paper. For both algorithms, the derived error exponents almost coincide with simulation results.