Unitary polylinear shift registers and their periods

In the article, a concept of a $k$-linear shift register ($k$-LSR) over a module ${}_RM$, where $R$ is an Artinian commutative ring, is studied. Such register is determined by a monic ideal $I\triangleleft R[x_1,\ldots,x_k]$ and a Ferrer diagram $\mathcal F\subset\mathbf N_0^k$. A class of ideals $I$ determining a $k$-LSR on some Ferrer diagram is described. In particular, a class of ideals $I$ determining a $k$-LSR on a fixed Ferrer diagram is constructed. A lower estimate for the periods of the constructed $k$-LSRs is obtained. It is shown that this estimate is attainable in some cases.