The decimation-Hadamard transform of two-level autocorrelation sequences

A new method to study and search for two-level autocorrelation sequences for both binary and nonbinary cases is developed. This method iteratively applies two operations: decimation and the Hadamard transform based on general orthogonal functions, referred to as the decimation-Hadamard transform (DHT). The second iterative DHT can transform one class of such sequences into another inequivalent class of such sequences, a process called realization. The existence and counting problems of the second iterative DHT are discussed. Using the second iterative DHT, and starting with a single binary m-sequence (when n is odd), we believe one can obtain all the known two-level autocorrelation sequences of period 2ⁿ-1 which have no subfield factorization. We have verified this for odd n⩽17. Interestingly, no previously unknown examples were found by this process for any odd n⩽17. This is supporting evidence (albeit weak) for the conjecture that all families of cyclic Hadamard difference sets of period 2ⁿ -1 having no subfield factorization are now known, at least for odd n. Experimental results are provided