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

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