Given a matrix
A
∈
G
L
d
(
Z
)
A\in \mathrm {GL}_d(\mathbb {Z})
. We study the pseudorandomness of vectors
u
n
\mathbf {u}_n
generated by a linear recurrence relation of the form
u
n
+
1
≡
A
u
n
(
mod
p
t
)
,
n
=
0
,
1
,
…
,
\begin{equation*} \mathbf {u}_{n+1} \equiv A \mathbf {u}_n \pmod {p^t}, \qquad n = 0, 1, \ldots , \end{equation*}
modulo
p
t
p^t
with a fixed prime
p
p
and sufficiently large integer
t
⩾
1
t \geqslant 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
\mathbf {u}_n
and control the
p
p
-adic orders of their coefficients in polynomial representations.