MOLS.bib

@Article{Dhaeseleer2017,
  author          = {D'haeseleer, Jozefien and Metsch, Klaus and Storme, Leo and
                  Van de Voorde, Geertrui},
  title           = {On the maximality of a set of mutually orthogonal Sudoku
                  Latin Squares},
  journal         = {Designs, Codes and Cryptography},
  year            = 2017,
  month           = {Jul},
  day             = 01,
  volume          = 84,
  number          = 1,
  pages           = {143-152},
  abstract        = {The maximum number of mutually orthogonal Sudoku Latin
                  squares (MOSLS) of order {\$}{\$}n=m^2{\$}{\$}n=m2is
                  {\$}{\$}n-m{\$}{\$}n-m. In this paper, we construct for
                  {\$}{\$}n=q^2{\$}{\$}n=q2, q a prime power, a set of
                  {\$}{\$}q^2-q-1{\$}{\$}q2-q-1MOSLS of order
                  {\$}{\$}q^2{\$}{\$}q2that cannot be extended to a set of
                  {\$}{\$}q^2-q{\$}{\$}q2-qMOSLS. This contrasts to the theory
                  of ordinary Latin squares of order n, where each set of
                  {\$}{\$}n-2{\$}{\$}n-2mutually orthogonal Latin Squares (MOLS)
                  can be extended to a set of {\$}{\$}n-1{\$}{\$}n-1MOLS (which
                  is best possible). For this proof, we construct a particular
                  maximal partial spread of size {\$}{\$}q^2-q+1{\$}{\$}q2-q+1in
                  {\$}{\$}{\backslash}mathrm {\{}PG{\}}(3,q){\$}{\$}PG(3,q)and
                  use a connection between Sudoku Latin squares and projective
                  geometry, established by Bailey, Cameron and Connelly.},
  issn            = {1573-7586},
  doi             = {10.1007/s10623-016-0234-3},
  url             = {https://doi.org/10.1007/s10623-016-0234-3}
}

@article{ETHIER2012430,
  title           = {Sets of orthogonal hypercubes of class r},
  journal         = {Journal of Combinatorial Theory, Series A},
  volume          = 119,
  number          = 2,
  pages           = {430-439},
  year            = 2012,
  issn            = {0097-3165},
  doi             = {https://doi.org/10.1016/j.jcta.2011.10.001},
  url             =
                  {https://www.sciencedirect.com/science/article/pii/S0097316511001609},
  author          = {John T. Ethier and Gary L. Mullen and Daniel Panario and
                  Brett Stevens and David Thomson},
  keywords        = {Hypercubes, Latin squares, Mutually orthogonal},
  abstract        = {A (d,n,r,t)-hypercube is an n×n×⋯×n (d-times) array on nr
                  symbols such that when fixing t coordinates of the hypercube
                  (and running across the remaining d−t coordinates) each symbol
                  is repeated nd−r−t times. We introduce a new parameter, r,
                  representing the class of the hypercube. When r=1, this
                  provides the usual definition of a hypercube and when d=2 and
                  r=t=1 these hypercubes are Latin squares. If d⩾2r, then the
                  notion of orthogonality is also inherited from the usual
                  definition of hypercubes. This work deals with constructions
                  of class r hypercubes and presents bounds on the number of
                  mutually orthogonal class r hypercubes. We also give
                  constructions of sets of mutually orthogonal hypercubes when n
                  is a prime power.}
}


@manual{GAP4,
  key             = "GAP",
  organization    = "The GAP~Group",
  title           = "{GAP -- Groups, Algorithms, and Programming, Version
                  4.11.0}",
  year            = 2020,
  url             = "https://www.gap-system.org",
}

@book {MR1644242,
  AUTHOR          = {Laywine, Charles F. and Mullen, Gary L.},
  TITLE           = {Discrete mathematics using {L}atin squares},
  SERIES          = {Wiley-Interscience Series in Discrete Mathematics and
                  Optimization},
  NOTE            = {A Wiley-Interscience Publication},
  PUBLISHER       = {John Wiley \& Sons, Inc., New York},
  YEAR            = 1998,
  PAGES           = {xviii+305},
  ISBN            = {0-471-24064-8},
  MRCLASS         = {05B15},
  MRNUMBER        = 1644242,
  MRREVIEWER      = {J. D\'{e}nes},
}

@article {MR1904388,
  AUTHOR          = {Morgan, Ilene H.},
  TITLE           = {Complete sets of mutually orthogonal hypercubes and their
                  connections to affine resolvable designs},
  JOURNAL         = {Ann. Comb.},
  FJOURNAL        = {Annals of Combinatorics},
  VOLUME          = 5,
  YEAR            = 2001,
  NUMBER          = 2,
  PAGES           = {227--240},
  ISSN            = {0218-0006},
  MRCLASS         = {05B30},
  MRNUMBER        = 1904388,
  MRREVIEWER      = {E. Seiden},
  DOI             = {10.1007/s00026-001-8009-5},
  URL             = {https://doi.org/10.1007/s00026-001-8009-5},
}

@inproceedings{minion06,
  author          = {Gent, Ian and Jefferson, Christopher and Miguel, Ian},
  year            = 2006,
  month           = 05,
  pages           = {98-102},
  title           = {Minion: A Fast Scalable Constraint Solver.},
  volume          = 2006,
  journal         = {Proceedings of ECAI 2006}
}