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Latin $p$-dimensional cube; Latin hypercube; Latin squares; orthogonal
We give a construction of $p$ orthogonal Latin $p$-dimensional cubes (or Latin hypercubes) of order $n$ for every natural number $n\ne 2,6$ and $p \ge 2$. Our result generalizes the well known result about orthogonal Latin squares published in 1960 by R. C. Bose, S. S. Shikhande and E. T. Parker.
[1] R. C. Bose, S. S. Shrikhande and E. T. Parker: Further results on the construction of mutually orthogonal Latin squares and the falsity of Euler’s conjecture. Canad. J.  Math. 12 (1960), 189–203. DOI 10.4153/CJM-1960-016-5 | MR 0122729
[2] J. Dénes and A. D. Keedwel: Latin Squares and Their Applications. Akadémiai Kiadó, Budapest, 1974. MR 0351850
[3] G. L. Mullen: Orthogonal hypercubes and related designs. J. Stat. Plann. Inference 73 (1998), 177–188. DOI 10.1016/S0378-3758(98)00059-7 | MR 1655219 | Zbl 0935.62089
[4] M. Trenkler: Magic $p$-dimensional cubes of order $n \lnot \equiv 2\hspace{4.44443pt}(\@mod \; 4)$. Acta Arithmetica 92 (2000), 189–194. MR 1750318
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