arXiv:2611.16299 · v1

Improved Lower Bounds for Mutually Orthogonal Latin Squares (MOLS) at 33 Orders

首次提交 2026-11-12 12:55

版本更新 2026-11-12 12:55

雷达观测 2026-11-13 14:16

摘要

A Latin square of order $n$ is an $n\times n$ array over symbols $\{0,1,\ldots n-1\}$, such that the same symbol does not appear twice in any row or any column. Two Latin squares of order $n$ are orthogonal if, when superimposed, they yield each of the $n^2$ ordered pairs of symbols exactly once. A set of two or more Latin squares are mutually orthogonal Latin squares (MOLS) if each pair of distinct squares is orthogonal. Let $N(n)$ be the maximum size of a set of MOLS of order $n$. For most $n$ which are not prime powers, there is a wide gap between known upper and lower bounds on $N(n)$. This has led to interest in constructing new larger sets of MOLS for explicit values of $n$, to incrementally improve the lower bounds on $N(n)$. This paper provides MOLS that improve the lower bound on $N(n)$ for 33 values of $n$ in the range from 22 through 494. The constructions were found by GPT-6 Astra, and use established techniques including difference matrices, quasi-difference matrices, orthogonal arrays, and projective planes. A highlight is 4 MOLS of order $n=22$; combined with pre-existing results, this establishes $N(n) \geq 4$ for all $n>10$.