Method for Transforming Matrix Circulants of Multiposition Linear Recurrence Sequences Into Matrices of Vilenkin-Crestenson Functions
A. V. Ermakova, S. F. Gorgadze
Abstract
The variants of constructing the circulant matrices of any p-ary linear recurrent sequence (LRS) of maximal period on the basis of automorphic multiplicative groups of the extended Galois field constructed by means of an irreducible primitive polynomial, on the basis of which the original p-ary LRS is formed, have been developed. As a result of this approach, new ways of transforming the circulant matrices of p-ary LRSs into the matrix of Vilenkin-Krestenson functions ordered by degrees of the primitive element of the field are revealed. It is shown for the first time that, depending on the initial conditions of the transformation, the set of arbitrary cyclic shifts of the p-ary LRS, shifted by one symbol relative to each other, can be transformed into arbitrary successive rows of the ordered matrix of Vilenkin-Krestenson functions. This fact makes it possible to simplify the algorithm of synchronisation of the p-ary LRS in a known range of its cyclic shifts, especially in the case of large periods of its repetition, and also to reduce the computational complexity of the processing algorithm when working in the truncated basis of the Vilenkin-Krestenson functions.