List decoding of biorthogonal codes and the Hadamard transform with linear complexity

Let a biorthogonal Reed-Muller code RM (1, m) of length n = 2(m) be used on a memoryless channel with an input alphabet +/- 1 and a real-valued output R. Given any nonzero received vector y in the Euclidean space R-n and some parameter epsilon epsilon (0, 1), our goal is to perform list decoding of the code RM and retrieve all codewords located within the angle arccos epsilon from y. For an arbitrarily small epsilon, we design an algorithm that outputs this list of codewords with the linear complexity order of n [1n(2) epsilon E] bit operations. Without loss of generality, let vector y be also scaled to the Euclidean length root n of the transmitted vectors. Then an equivalent task is to retrieve all coefficients of the Hadamard transform of vector y whose absolute values exceed n epsilon. Thus, this decoding algorithm retrieves all n epsilon-significant coefficients of the Hadamard transform with the linear complexity n [1n(2) epsilon] instead of the complexity n1n(2) n of the full Hadamard transform.