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, [formula presented] [formula presented] is examined analytically in the thermodynamic limit where the main results are (a) For the binary case, the minimal value of [formula presented] over all sequences of length N, [formula presented] is [formula presented] significantly smaller than the typical value for random sequences [formula presented] (b) A new method to approximate [formula presented] is obtained using the observation of data collapse. (c) [formula presented] is obtained in an energy which is about [formula presented] above the ground-state energy of the generalized Bernasconi model, independent of the number of phases m. (d) For a given m, [formula presented] indicating that for [formula presented] [formula presented] i.e., a generalized Barker code exists. The analytical results are confirmed by simulations.