Distribution of recursive matrix pseudorandom number generator modulo prime powers

Given a matrix A is an element of GL(d)(Z). We study the pseudorandomness of vectors un generated by a linear recurrence relation of the form u(n+1) = Au-n (mod p(t)), n = 0,1, ... , modulo p(t) with a fixed prime p and sufficiently large integer t >= 1. We study such sequences over very short segments of length which has not been accessible via previously used methods. Our technique is based on the method of N. M. Korobov [Mat. Sb. (N.S.) 89(131) (1972), pp. 654-670, 672] of estimating double Weyl sums and a fully explicit form of the Vinogradov mean value theorem due to K. Ford [Proc. London Math. Soc. (3) 85 (2002), pp. 565- 633]. This is combined with some ideas from the work of I. E. Shparlinski [Proc. Voronezh State Pedagogical Inst., 197 (1978), 74-85 (in Russian)] which allows us to construct polynomial representations of the coordinates of u(n) and control the p-adic orders of their coefficients in polynomial representations.