Group divisible designs in MOLS of order ten - Dukes, Peter; Howard, Leah (Designs, codes, and cryptography, 2014)
The maximum number of mutually orthogonal latin squares (MOLS) of order 10 is known to be between 2 and 6. A hypothetical set of four MOLS must contain at least one of the types of group divisible designs (GDDs) classified here. The proof is based on a dimension argument modified from work by Dougherty. The argument has recently led to the discovery of a counterexample to Moorhouse’s conjecture on the rank of nets, found by Howard and Myrvold. Although it is known that even three MOLS can admit no nontrivial symmetry group, we are hopeful this classification via GDDs and dimension can offer some structure to aid the eventual goal of exhausting the search for four MOLS of order 10.