A lower bound on HMOLS with equal sized holes

TL;DR

This paper establishes a new lower bound on the maximum number of mutually orthogonal Latin squares with a common equipartition into holes, extending previous results and using advanced combinatorial and number theory techniques.

Contribution

It generalizes the difference matrix method and applies cyclotomic number estimates to derive a lower bound for HMOLS with equal-sized holes.

Findings

Established lower bound N(h^n) ≥ (log n)^{1/δ} for HMOLS

Extended difference matrix construction techniques

Applied cyclotomic number estimates to combinatorial design

Abstract

It is known that $N(n)$, the maximum number of mutually orthogonal latin squares of order $n$, satisfies the lower bound $N(n) \ge n^{1/14.8}$ for large $n$. For $h\ge 2$, relatively little is known about the quantity $N(h^n)$, which denotes the maximum number of `HMOLS' or mutually orthogonal latin squares having a common equipartition into $n$ holes of a fixed size $h$. We generalize a difference matrix method that had been used previously for explicit constructions of HMOLS. An estimate of R.M. Wilson on higher cyclotomic numbers guarantees our construction succeeds in suitably large finite fields. Feeding this into a generalized product construction, we are able to establish the lower bound $N(h^n) \ge (\log n)^{1/\delta}$ for any $\delta>2$ and all $n > n_0(h,\delta)$.