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Figure: A 6 x 6 x 6 Costas cube. The Costas cube is a newly devised mathematical structure that contains 216 cells of size 1 x 1 x 1, six of which are shown here as red cells. Each of the three projections of the cube onto two dimensions (from below, from the back, and from the side) is a 6 x 6 Costas array in which the 1x1 shadows of the six red cells on the two-dimensional surface reveal the six positions of the square grid that should be coloured dark to form a Costas array.

Shining a light on the existence pattern for Costas arrays

The motivation Costas arrays are square grids of cells each coloured dark or light so that the pattern of dark cells within the grid satisfies certain mathematical rules. Proposed in the 1960s, this type of an array was developed as a mathematical approach to improving the performance of the radar and sonar systems in existence at the time. Since then, the principal mathematical objective of research into Costas arrays has been to determine all the sizes of square grids for which these arrays exist. While brainstorming possible new ways to construct Costas arrays, Simon Fraser University mathematicians Jonathan Jedwab and Lily Yen stumbled upon the idea of a three-dimensional structure that encodes three 2-D Costas arrays simultaneously.

The discovery – Drs. Jedwab and Yen discovered infinitely many three-dimensional configurations of cells that they called Costas cubes. If you shine a light on the cube from below, the back, or the side, then the shadow cast by the cube  – in all three cases – is a Costas array. This discovery was particularly exciting, as it wasn’t clear at the outset that it would indeed be possible to find any such three-dimensional configurations.  

Its significance – The last major discovery of a construction for Costas arrays occurred almost 35 years ago. Although the researchers have not yet generated new Costas arrays from the Costas cubes, the discovery of these cubes is significant because they introduce an original geometric point of view and provide new perspectives on the observed existence pattern for Costas arrays. Looking forward, Costas cubes have possible applications in digital communications, such as optical orthogonal codes (which improve the efficiency and reliability of information transmission over networks of optical fibers) and digital watermarking (which covertly identify the copyright owner of audio, video or image data).

Read the paper“Costas cubes” by Jonathan Jedwab and Lily Yen. IEEE Transactions on Information Theory 64(4):3144-3149 (2018). DOI: 10.1109/TIT.2017.2763962

Website article compiled by Jacqueline Watson with Theresa Kitos