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The de Bruijn torus (or grid) problem looks to find an $n$-by-$m$ binarymatrix in which every possible $j$-by-$k$ submatrix appears exactly once. Theexistence 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, theseconstructions lacked efficient decoding methods, leading to new constructionsin the early 2000's. The new constructions develop cross-shaped patterns(rather than rectangular), and rely on a concept known as a half de Bruijnsequence. In this paper, we further advance this construction beyondcross-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 asrobotic vision, location detection, and projective touch-screen displays.

Locating Patterns in the De Bruijn Torus