Correlational properties of the full-length sequences based on the discretized Markov β-transformations

Abstract

We have previously defined the discretized Markov transformations and the full-length sequences based on such transformations. In view of basic properties of the normalized cross- and auto-correlation functions for the de Bruijn sequences that can be regarded as the full-length sequences based on the discretized dyadic transformation, we obtain correlational properties of the full-length sequences based on the discretized golden mean transformation. We generalize this result and give the correlational properties of the discretized Markov β-transformations with the alphabet Σ={0,1,···,k-1} and the set F={(k-1)(k-1)} of forbidden blocks (k≥2), whose underlying transformations exhibit the most fundamental class of greedy β-expansions of real numbers. We also apply the generalized result to evaluate the auto-correlation function for the optimum binary spreading sequences of Markov chains based on discretized β-transformations.