A Census of Small Latin Hypercubes

We count all latin cubes of order n 6 and latin hypercubes of order n 5 and dimension d 5. We classify these (hyper)cubes into isotopy classes and paratopy classes (main classes). For the same values of n and d we classify all d-ary quasigroups of order n into isomorphism classes and also count them according to the number of identity elements they possess (meaning we have counted the d-ary loops). We also give an exact formula for the number of (isomorphism classes of) d-ary quasigroups of order 3 for every d. Then we give a number of constructions for d-ary quasigroups with a specific number of identity elements. In the process, we prove that no 3-ary loop of order n can have exactly n-1 identity elements (but no such result holds in dimensions other than 3). Finally, we give some new examples of latin cuboids which cannot be extended to latin cubes. 1 Basic definitions Let [n] denote the set {1, 2, . . . , n} and let [n]d denote the cartesian product [n]

• Type: Journal
• Year: 2008
• Where: SIAMDM
• Authors: Brendan D. McKay, Ian M. Wanless
• Keywords: D-ary Quasigroups, Hypercube, Isomorphism Classes, SIAMDM 2008