Enumeration of MOLS of small order

We report the results of a computer investigation of sets of mutually orthogonal latin squares (MOLS) of small order. For $n\le9$ we 1. Determine the number of orthogonal mates for each species of latin square of order $n$. 2. Calculate the proportion of latin squares of order $n$ that have an orthogonal mate, and the expected number of mates when a square is chosen uniformly at random. 3. Classify all sets of MOLS of order $n$ up to various different notions of equivalence. We also provide a triple of latin squares of order 10 that is the closest to being a set of MOLS so far found.

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

[2]  H. W. Norton.  THE 7 × 7 SQUARES , 1939 .

[3]  David A. Drake,et al.  Maximal sets of Latin squares and partial transversals , 1977 .

[4]  C. Colbourn,et al.  Handbook of Combinatorial Designs , 2006 .

[5]  David A. Drake,et al.  The Non-Existence of Maximal Sets of Four Mutually Orthogonal Latin Squares of Order 8 , 2004, Des. Codes Cryptogr..

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

[7]  Anthony B. Evans.  Latin Squares without Orthogonal Mates , 2006, Des. Codes Cryptogr..

[8]  Mike J. Grannell,et al.  On the number of transversal designs , 2013, J. Comb. Theory A.

[9]  P. J. Owens,et al.  Complete Sets of Pairwise Orthogonal Latin Squares and the Corresponding Projective Planes , 1992, J. Comb. Theory, Ser. A.

[10]  Ian M. Wanless.  Perfect Factorisations of Bipartite Graphs and Latin Squares Without Proper Subrectangles , 1999, Electron. J. Comb..

[11]  Patric R. J. Östergård.  Constructing combinatorial objects via cliques , 2005, Surveys in Combinatorics.

[12]  K. Kunen.  The structure of conjugacy closed loops , 2000 .

[13]  Brendan D. McKay,et al.  The number of transversals in a Latin square , 2006, Des. Codes Cryptogr..

[14]  Brendan D. McKay,et al.  Practical graph isomorphism, II , 2013, J. Symb. Comput..

[15]  Ian M. Wanless,et al.  A Generalisation of Transversals for Latin Squares , 2002, 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]  Patric R. J. Östergård,et al.  The number of Latin squares of order 11 , 2009, Math. Comput..

[19]  E. T. Parker,et al.  The existence and construction of two families of designs for two successive experiments , 1970 .

[20]  Ian M. Wanless,et al.  The Existence of Latin Squares without Orthogonal Mates , 2006, Des. Codes Cryptogr..

[21]  Dieter Jungnickel,et al.  Maximal difference matrices of order <=10 , 1986, Discret. Math..

[22]  Ian M. Wanless,et al.  Monogamous latin squares , 2011, J. Comb. Theory, Ser. A.

[23]  Ian M. Wanless,et al.  Atomic Latin Squares of Order Eleven , 2004 .

[24]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[25]  P. J. Owens,et al.  Aspects of complete sets of 9 × 9 pairwise orthogonal latin squares , 1997, Discret. Math..

[26]  E. H. Moore,et al.  Tactical Memoranda I-III , 1896 .

[27]  Ian M. Wanless,et al.  Erratum: Latin squares with restricted transversals (Journal of Combinatorial Designs (2012) 20 (124-141)) , 2012 .

[28]  I. Wanless,et al.  Indivisible partitions of latin squares , 2011 .

[29]  P. Kaski,et al.  Classification Algorithms for Codes and Designs , 2005 .

[30]  Ian M. Wanless,et al.  Latin Squares with Restricted Transversals , 2012 .