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 plusmn1 and a real-valued output R. Given any nonzero received vector y in the Euclidean space Rⁿ and some parameter epsiisin(0,1), our goal is to perform list decoding of the code RM (1, m) and retrieve all codewords located within the angle arccos e from y. For an arbitrarily small epsi, we design an algorithm that outputs this list of codewords with the linear complexity order of n [ln² isin] bit operations. Without loss of generality, let vector y be also scaled to the Euclidean length radic(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 nisin. Thus, this decoding algorithm retrieves all ne-significant coefficients of the Hadamard transform with the linear complexity n [ln² isin] instead of the complexity n In²n of the full Hadamard transform.