Error Exponents for Two Soft-Decision Decoding Algorithms of Reed–Muller Codes

Error exponents are studied for recursive and majority decoding of general Reed–Muller (RM) codes ${rm RM}(r,m)$ used on the additive white Gaussian noise (AWGN) channels. Both algorithms have low complexity and correct many error patterns whose weight exceeds half the code distance. Decoding consists of multiple consecutive steps, which repeatedly recalculate the input symbols and determine different information symbols using soft-decision majority voting. For any code ${rm RM}(r,m)$, we estimate the probabilities of the information symbols obtained in these recalculations and derive the analytical upper bounds for the block error rates of the recursive and majority decoding. In the case of a low noise, we also obtain the lower bounds and show that the upper bounds are tight. For a higher noise, these bounds closely approach our simulation results.