Existence of Universal Functions for the Class of Linear k-Valued Functions with Moderate k

Abstract

The article describes the construction of discrete functions which, by some of their values, specify (generate) arbitrary linear functions. The cases of prime and sufficiently large composite k have been considered previously. In the present study we finally solve the problem of existence of such functions for almost all k and n variables. The proof of the probabilistic upper bound and the general approach are due to A. A. Voronenko. The proof for small k has been developed by N. K. Voronova. The proof for k from 21 to 48 is the result of indispensable cooperation of V. P. Il’yutko and A. A. Voronenko.