32 research outputs found
Enumerating extensions of mutually orthogonal Latin squares
Two nĂ—n Latin squares L1,L2 are said to be orthogonal if, for every ordered pair (x, y) of symbols, there are coordinates (i, j) such that L1(i,j)=x and L2(i,j)=y. A k-MOLS is a sequence of k pairwise-orthogonal Latin squares, and the existence and enumeration of these objects has attracted a great deal of attention. Recent work of Keevash and Luria provides, for all fixed k, log-asymptotically tight bounds on the number of k-MOLS. To study the situation when k grows with n, we bound the number of ways a k-MOLS can be extended to a (k+1)-MOLS. These bounds are again tight for constant k, and allow us to deduce upper bounds on the total number of k-MOLS for all k. These bounds are close to tight even for k linear in n, and readily generalise to the broader class of gerechte designs, which include Sudoku squares
Pairwise balanced designs covered by bounded flats
We prove that for any and , there exist, for all sufficiently large
admissible , a pairwise balanced design PBD of dimension for
which all -point-generated flats are bounded by a constant independent of
. We also tighten a prior upper bound for , in which case
there are no divisibility restrictions on the number of points. One consequence
of this latter result is the construction of latin squares `covered' by small
subsquares