Soft-decision list decoding of Reed-Muller codes with linear complexity

Let a binary Reed-Muller code RM(s;m) of length n be used on a memoryless channel with an input alphabet ±1 and a real-valued output ℝ. Given a received vector y in ℝ_n; we define its generalized distance T to any codeword c as the sum Σ|y_j| taken over all positions j, in which vectors y, c have opposite signs. We then consider the list ℒ_T of codewords located within distance T from the received vector y and estimate the size L_T of this list using the generalized Johnson bound. For any RM code RM(s,m) of fixed order s, the algorithm is proposed that performs list decoding beyond the error-correcting radius with linear complexity in length n and retrieves the code list ℒ_T with complexity of order nsL_T for any decoding radius T within the generalized Johnson bound.