De Bruijn torus

Apart from "translation", "inversion" (exchanging 0s and 1s) and "rotation" (by 90 degrees), no other (4,4;2,2)2 de Bruijn tori are possible – this can be shown by complete inspection of all 216 binary matrices (or subset fulfilling constrains such as equal numbers of 0s and 1s).

All n×n submatrices without wraparound, such as the one shaded yellow, then form the complete set: An example of the next possible binary "square" de Bruijn torus, (256,256;4,4)2 (abbreviated as B4), has been explicitly constructed.

The paper in which an example of the (256,256;4,4)2 de Bruijn torus was constructed contained over 10 pages of binary, despite its reduced font size, requiring three lines per row of array.

The object B8, containing all binary 8×8 matrices and denoted (4294967296,4294967296;8,8)2, has a total of 264 ≈ 18.447×1018 entries: storing such a matrix would require 18.5 exabits, or 2.3 exabytes of storage.

De Bruijn tori are used in the spatial coding context, e.g. for localization of a camera,[6] a robot[7] or a tangible[8] based on some optical ground pattern.

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) ​
The (4,4;2,2) de Bruijn torus. Each 2-by-2 binary matrix can be found within it exactly once.
De Bruijn torus (8,8;3,2) containing all 64 possible 3-row × 2-column matrices exactly once, with wrap­around – the bottom half is the negative of the top half
B 4 as a binary square matrix
The grid highlights some of the 4×4 matrices, including those of zeros and of ones at the upper margin.
Simplified principle of the Anoto digital pen.
The camera identifies a 6×6 matrix of dots, each displaced from the blue grid (not printed) in one of 4 directions.
The combinations of relative displacements of a 6-bit de Bruijn sequence between the columns, and between the rows gives its absolute position on the digital paper.
An example of a PuzzleBoard pattern with 8x11 chessboard corners.
An example of a PuzzleBoard pattern with 8x11 chessboard corners. Each 3x3 tile pattern is unique.