Theoretical Foundations of Applied Discrete Mathematics

Digit-polynomial construction of substitutions over Galois ring

M. V. Zaets

Research Institute "Kvant", Moscow

Abstract: A new way for constructing substitutions over Galois ring is considered. The way uses functions with variational-digit polynomiality. The class of these functions over various rings was earlier defined in the author's works. The peculiarity of this class is that it contains a class of polynomial functions and, under certain conditions, does not coincide with it. The criterions for the bijectivity of a polynomial vector-function and for a polynomial function to be a substitution are generalized. The presented results make it possible, in particular, to construct non-polynomial $n$-quasigroups.

Keywords: substitutions, $n$-quasigroups, bijective polynomial vector-function, functions with variational-digit polynomiality, digit set, Galois ring.

UDC: 519.716.32+519.854

DOI: 10.17223/2226308X/10/5