List Decoding of Biorthogonal Codes and the Hadamard Transform With Linear Complexity

length n =2 m be used on a memoryless channel with an input alphabet 61 and a real-valued output. Given any nonzero received vector y in the Euclidean space n and some parameter 2 (0; 1), our goal is to perform list decoding of the code RM (1;m) and retrieve all codewords located within the angle arccos from y. For an arbitrarily small, we design an algorithm that outputs this list of codewords with the linear complexity order of n ln 2 bit operations. Without loss of generality, let vector y be also scaled to the Euclidean length p 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. Thus, this decoding algorithm retrieves all n-significant coefficients of the Hadamard transform with the linear complexity n ln 2 instead of the complexity n ln 2 n of the full Hadamard transform