Soft-Decision List Decoding with Linear Complexity for the First-Order Reed-Muller Codes

Soft-decision decoding on a memoryless channel is considered for the first-order Reed-Muller codes RM(1,m) of length 2^m. We assume that different positions j of the received binary vector y can be corrupted by the errors of varying weight W_j. The generalized Hamming distance between vector y and any binary vector c is then defined as the sum of weighted differences W_j|y_j - C_j| taken over all n positions. We obtain a tight upper bound Lt on the number of codewords located within generalized Hamming distance T from vector y, and design a decoding algorithm that outputs this list of codewords with complexity O(n In² Lt)- In particular, all possible error weights w_j equal 1 if this combinatorial model is applied to a binary symmetric channel. In this case, the well known Green algorithm performs full maximum likelihood decoding of RM(1,m) and requires O(n ln² n) bit operations, whereas the Litsyn-Shekhovtsov algorithm operates within the bounded-distance decoding radius n/4 - 1 with linear complexity O(n). We close the performance-complexity gap between the two algorithms. Namely, for any fixed epsi isin (0, frac12), our algorithm outputs the complete list of codewords within the decoding radius n(frac12 - epsi) with linear complexity of order n ln² epsi.