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arXiv · 2001.05767

Universal arrays

Matías Pavez-Signé·Daniel A. Quiroz·Nicolás Sanhueza-Matamala
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Abstract

A word on $q$ symbols is a sequence of letters from a fixed alphabet of size $q$. For an integer $k\ge 1$, we say that a word $w$ is $k$-universal if, given an arbitrary word of length $k$, one can obtain it by removing entries from $w$. It is easily seen that the minimum length of a $k$-universal word on $q$ symbols is exactly $qk$. We prove that almost every word of size $(1+o(1))c_qk$ is $k$-universal with high probability, where $c_q$ is an explicit constant whose value is roughly $q\log q$. Moreover, we show that the $k$-universality property for uniformly chosen words exhibits a sharp threshold. Finally, by extending techniques of Alon [Geometric and Functional Analysis 27 (2017), no. 1, 1--32], we give asymptotically tight bounds for every higher dimensional analogue of this problem.

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BibTeXRIS

Matías Pavez-Signé, Daniel A. Quiroz, Nicolás Sanhueza-Matamala. 2020-01-16. Universal arrays. https://doi.org/10.1016/j.disc.2021.112626

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