The interplay between the ground state energy of the generalized Bernasconi model to multi-phase, and the minimal value of the maximal autocorrelation function, $C_{ max}=\max_K{ |C_K|}$, $K=1,..N-1$, is examined analytically and the main results are: (a) The minimal value of $\min_N{ C_{ max}}$ is $0.435\sqrt{ N}$ significantly smaller than the typical value for random sequences $O(\sqrt{ \log{ N}}\sqrt{ N})$. (b) $\min_N{ C_{ max}}$ over all sequences of length N 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. (c) The maximal merit factor $F_{ max}$ grows linearly with m. (d) For a given N, $\min_N{ C_{ max}}\sim\sqrt{ N/m}$ indicating that for m=N, $\min_N{ C_{ max}}=1$, i.e. a Barker code exits. The analytical results are confirmed by simulations.
Categories: cond-mat.stat-mech cond-mat.dis-nn