  The de Bruijn torus (or grid) problem looks to find an $n$-by-$m$ binary
matrix in which every possible $j$-by-$k$ submatrix appears exactly once. The
existence and construction of these binary matrices was determined in the 70's,
with generalizations to $d$-ary matrices in the 80's and 90's. However, these
constructions lacked efficient decoding methods, leading to new constructions
in the early 2000's. The new constructions develop cross-shaped patterns
(rather than rectangular), and rely on a concept known as a half de Bruijn
sequence. In this paper, we further advance this construction beyond
cross-shape patterns. Furthermore, we show results for universal cycle grids,
based off of the one-dimensional universal cycles introduced by Chung,
Diaconis, and Graham, in the 90's. These grids have many applications such as
robotic vision, location detection, and projective touch-screen displays.
