Low autocorrelated multiphase sequences

The interplay between the ground-state energy of the generalized Bernasconi model to multiphase, and the minimal value of the maximal autocorrelation function, C_max=max_K\|C_K\|, K=1,...,N-1, is examined analytically in the thermodynamic limit where the main results are (a) For the binary case, the minimal value of C_max over all sequences of length N, minC_max, is 0.435(N), significantly smaller than the typical value for random sequences O((log N)(N)). (b) A new method to approximate F_max is obtained using the observation of data collapse. (c) minC_max is obtained in an energy which is about 30% above the ground-state energy of the generalized Bernasconi model, independent of the number of phases m. (d) For a given m, minC_max~(N/m) indicating that for m=N, minC_max=1, i.e., a generalized Barker code exists. The analytical results are confirmed by simulations.