Mutually orthogonal latin squares with large holes

TL;DR

This paper proves the existence of large sets of incomplete mutually orthogonal latin squares with holes, establishing bounds on their order relative to the size of the holes and the number of squares.

Contribution

It provides a construction for incomplete mutually orthogonal latin squares with large holes, extending the theory to cases where the order is significantly larger than the hole size.

Findings

Existence of incomplete mutually orthogonal latin squares for all large n,m satisfying n ≥ 8(t+1)^2 m.

Establishment of bounds relating the order of the squares, the number of squares, and the size of the holes.

Abstract

Two latin squares are orthogonal if, when they are superimposed, every ordered pair of symbols appears exactly once. This definition extends naturally to `incomplete' latin squares each having a hole on the same rows, columns, and symbols. If an incomplete latin square of order $n$ has a hole of order $m$, then it is an easy observation that $n \ge 2m$. More generally, if a set of $t$ incomplete mutually orthogonal latin squares of order $n$ have a common hole of order $m$, then $n \ge (t+1)m$. In this article, we prove such sets of incomplete squares exist for all $n,m \gg 0$ satisfying $n \ge 8(t+1)^2 m$.