Array containing every possible matrix of size m × n
STL model of de Bruijn torus (16,32;3,3)2 with 1s as panels and 0s as holes in the mesh – with consistent orientation, every 3×3 matrix appears exactly once (external viewer)
In combinatorial mathematics, a De Bruijn torus, named after Dutch mathematician Nicolaas Govert de Bruijn, is an array of symbols from an alphabet (often just 0 and 1) that contains every possible matrix of given dimensions m × n exactly once. It is a torus because the edges are considered wraparound for the purpose of finding matrices. Its name comes from the De Bruijn sequence, which can be considered a special case where n = 1 (one dimension).
One of the main open questions regarding De Bruijn tori is whether a De Bruijn torus for a particular alphabet size can be constructed for a given m and n. It is known that these always exist when n = 1, since then we simply get the De Bruijn sequences, which always exist. It is also known that "square" tori exist whenever m = n and even (for the odd case the resulting tori cannot be square).[1][2][3]
The smallest possible binary "square" de Bruijn torus, depicted above right, denoted as (4,4;2,2)2 de Bruijn torus (or simply as B2), contains all 2×2 binary matrices.
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Fan, C. T.; Fan, S. M.; Ma, S. L.; Siu, M. K. (1985). "On de Bruijn arrays". Ars Combinatoria A. 19: 205–213.
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Chung, F.; Diaconis, P.; Graham, R. (1992). "Universal cycles for combinatorial structures". Discrete Mathematics. 110 (1): 43–59. doi:10.1016/0012-365x(92)90699-g.
The smallest possible binary "square" deBruijntorus, depicted above right, denoted as (4,4;2,2)2 deBruijntorus (or simply as B2), contains all 2×2 binary...
on the deBruijntorus. Computing the position of a particular unique tuple or matrix in a deBruijn sequence or torus is known as the deBruijn decoding...
mathematics, a deBruijntorus is an array of symbols from an alphabet (often just 0 and 1) that contains every m-by-n matrix exactly once. It is a torus because...
Nicolaas Govert "Dick" deBruijn (Dutch: [nikoːˈlaːs ˈxoːvərt də ˈbrœyn]; 9 July 1918 – 17 February 2012) was a Dutch mathematician, noted for his many...
subsequence appears exactly once DeBruijntorus, a generalization of the DeBruijn sequence in two dimensions DeBruijn graph, a graph representing overlaps...
solved by the Gale–Ryser theorem. List of matrices Binatorix (a binary DeBruijntorus) Bit array Disjunct matrix Redheffer matrix Truth table Three-valued...
interconnection network such as a deBruijn graph, a hypercube graph, a hypertree network, a fat tree network, a torus, or cube-connected cycles. A grid...
example, the torus has Euler characteristic χ = 0 (and genus g = 1) and thus p = 7, so no more than 7 colors are required to color any map on a torus. This upper...
Hualde, Jon Ortiz De Urbina, Walter de Gruyter, 2003 Ph Bloemhoff-deBruijn, Anderhalve Eeuw Zwols Vocaalveranderingsprocessen in de periode 1838-1972...
most 3, while the queue number of planar 3-trees is at most 5. Binary deBruijn graphs have queue number 2. The d-dimensional hypercube graph has queue...
Monique Bouwman, Jan Braun, Marie Brienesse, Karin Brigitha, Enith Bruijn, Inge de Bruins, Rika Brzoskowski, Maarten Bulten, Kira Bunschoten, Hansje Buter...
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