De Bruijn环面中的定位模式
Locating Patterns in the De Bruijn Torus
摘要：
de Bruijn环面（或网格）问题寻找一个$n$乘$m$的二进制矩阵，其中每个可能的$j$乘$k$子矩阵只出现一次。这些二元矩阵的存在性和构造在70年代被确定，在80年代和90年代被推广到$d$ -元矩阵。然而，这些构造缺乏有效的解码方法，导致21世纪初的新构造。新的结构发展了十字形图案（而不是矩形），并依赖于一个被称为半德布鲁恩序列的概念。在本文中，我们进一步推进这种结构超越十字形模式。此外，我们显示了通用循环网格的结果，基于Chung，Diaconis和Graham在90年代引入的一维通用循环。这些网格有许多应用，如机器人视觉，位置检测和投影触摸屏显示。
英文摘要：
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.
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