On the outcome of false images of linear k-valued functions for composite numbers k when the number of variables increases

The problem of constructing discrete functions such that parts of their value sets determine (generate) arbitrary linear functions is considered. A case in which k is a prime number was considered earlier by the author. It is proved that the existence of such partial functions wshen the number of independent variables is no less then two implies they exists for any arbitrary greater number of independent variables. Upper estimates linear with respect to the number of independent variables are proved for the size of the domain of universal functions. The existence of two-variable universal functions is proved for sufficiently large k .