On the Statistical Independence of Shift-Register Pseudorandom Multisequence over Part of the Period

Abstract

In this paper we construct a pseudorandom multisequence $(x_{n_1,...,n_r})$ based on $k$th-order linear recurrences modulo $p$, such that the discrepancy of the $s$-dimensional multisequence $(x_{n_1+i_1,...,n_r+i_r}{)}_{1\le i_j\le s_j,1\le j\le r}$ $1\le n_j\le N_j,1\le j\le r$ is equal to $O((N_1...N_r{)}^{-1/2}{\ln }^{s+3r}(N_1...N_r))$, where $s=s_1...s_r$, for all $N_1,...,N_r$ with $1 < N_1 ... N_r \leq p^k