A Census of Small Latin Hypercubes

We count all latin cubes of order $n\le6$ and latin hypercubes of order $n\le5$ and dimension $d\le5$. 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]  Victor Zinoviev,et al.  Binary Extended Perfect Codes of Length 16 by the Generalized Concatenated Construction , 2002, Probl. Inf. Transm..

[2]  M. Hall.  An existence theorem for latin squares , 1945 .

[3]  Allan B. Cruse.  On the Finite Completion of Partial Latin Cubes , 1974, J. Comb. Theory, Ser. A.

[4]  Luc Teirlinck.  Generalized idempotent orthogonal arrays , 1990 .

[5]  H. Wilf.  generatingfunctionology: Third Edition , 1990 .

[6]  J. Dénes,et al.  Latin squares and their applications , 1974 .

[7]  S. C. Pearce,et al.  Orthogonal designs for three-dimensional experiments , 1973 .

[8]  Gary L. Mullen,et al.  LATIN CUBES AND HYPERCUBES OF PRIME ORDER , 1985 .

[9]  Gary L. Mullen,et al.  Latin cubes of order >=5 , 1980, Discret. Math..

[10]  Ian M. Wanless,et al.  On the Number of Latin Squares , 2005, 0909.2101.

[11]  N. J. A. Sloane,et al.  The On-Line Encyclopedia of Integer Sequences , 2003, Electron. J. Comb..

[12]  Charles C. Lindner.  A Finite Partial Idempotent Latin Cube Can Be Embedded in a Finite Idempotent Latin Cube , 1976, J. Comb. Theory, Ser. A.

[13]  Martin Kochol,et al.  Relatively narrow latin parallelpipeds that cannot be extended to a latin cube , 1995, Ars Comb..

[14]  Denis Krotov,et al.  Asymptotics for the number of n-quasigroups of order 4 , 2006 .

[15]  Jonathan Cutler,et al.  Latin Squares with Forbidden Entries , 2006, Electron. J. Comb..

[16]  G. Mullen,et al.  Discrete Mathematics Using Latin Squares , 1998, The Mathematical Gazette.

[17]  B. McKay,et al.  Small latin squares, quasigroups, and loops , 2007 .

[18]  Katherine Heinrich.  Prolongation in m-Dimensional Permutation Cubes , 1982 .