Possible sets of autocorrelations and the Simplex algorithm

The problem of imposing a set of correlations, , of any order, , on binary sequences is addressed. The entropy of infinitely long sequences obeying such a given set was calculated in previous works using the saddle-point method, and it was observed that a finite fraction of sets are characterized by a non-extensive entropy. In this paper, the region of finite entropy, the allowed region of sets of correlations, is found to be a convex hyper-polygon in the space of correlation-sets, using the Simplex algorithm. Outside of this region the Simplex solution indicates that sequences obeying the correlations cannot be found; therefore, the entropy is -∞. In particular, the boundaries of the allowed region for {C₁, C_m} are presented. At the boundaries, the entropy drops in a first-order phase transition fashion, and this drop can be explained from a combinatorial point of view. Finally, we observe that the fraction of the volume occupied by allowed correlation-sets drops exponentially with the number of correlations imposed, and a qualitative explanation of this scaling phenomenon is provided.