Omnimosaics

TL;DR

This paper introduces the concept of omnimosaics, a special type of matrix containing all possible submatrices of a given size, and provides bounds on the minimal size needed for such matrices as the submatrix size grows.

Contribution

The paper defines omnimosaics and establishes asymptotic bounds on their minimal size for fixed alphabet size, advancing combinatorial matrix construction knowledge.

Findings

Established bounds on the size of omnimosaics as submatrix size increases

Provided explicit constructions for omnimosaics

Demonstrated asymptotic behavior of minimal omnimosaic size

Abstract

An {\it omnimosaic} $O(n,k,a)$ is defined to be an $n\times n$ matrix, with entries from the set ${\cal A}=\{1,2,\...,a\}$, that contains, as a submatrix, each of the $a^{k^2}$ $k\times k$ matrices over ${\cal A}$. We provide constructions of omnimosaics and show that for fixed $a$ the smallest possible size $\omega(k,a)$ of an $O(n,k,a)$ omnimosaic satisfies \[\frac{ka^{k/2}}{e}\le \omega(k,a)\le \frac{ka^{k/2}}{e}(1+o(1))\] for a well-specified function $o(1)$ that tends to zero as $k\to\infty$.