torus packing for multisets Software

This project contains Python3 code that illustrates the algorithms from the paper "Robot positioning using torus packing for multisets" by (in alphabetical order) Chung Shue Chen (Nokia Bell Labs, France) Élie de Panafieu (Nokia Bell Labs, France) Peter Keevash, (Mathematical Institute, University of Oxford, UK) Sean Kennedy (Nokia Bell Labs, Canada) Adrian Vetta (McGill University, Canada) presented at the International Colloquium on Automata, Languages and Programming, ICALP Track A 2024. In the terminology of "Universal cycles for combinatorial structures" (Fan Chung, Persi Diaconis, Ron Graham, 1992) and "Universal Cycle Packings and Coverings for k-Subsets of an n-Set" (Michał Dȩbski, Zbigniew Lonc, 2016), our algorithm outputs a universal cycle packing of dimension d for multisets. Let us now detail what it means. It inputs integer parameters 'd', 'b' and 't' and outputs a colored grid. Let k = db + 1, m = 2bdt, s = (2bd)(d-2) * t(d-1) and n = (2ms + 1)**(b-1) * (2ms - 1/s) + 1/s. The grid has dimension 'd' and size 'n'. Each pixel receives a color in {0, 1, ..., k-1}. The grid is considered as a torus. The main property is that no two d-dimensional subsquare of size m correspond to the same multiset of colors.