Representation and simulating of extended Markov chains over a finite field with a predefined precision

Abstract

A method is proposed for representing and simulating the extended Markov chains of a defined type by minimal polynomials over finite field GF(q). Simulation problem is being solved as a problem of constructing the minimal polynomial over field GF(q) using the Berlekamp-Massey algorithm. The polynomial produces a sequence of the length of N. A stochastic matrix relevant to that sequence approximates the given stochastic matrix of an extended Markov chain with a predefined precision proportional to the 1/N value. The polynomial constructed defines unambiguously the structure of a q-ary linear shift register for simulating extended Markov chains. The method allows constructing the implementations of this class of Markov sequences on linear q-ary shift registers with a pre-defined “linear complexity” defined by the value of N.