Explicit Orthogonal Arrays and Universal Hashing with Arbitrary Parameters
Abstract
Orthogonal arrays are a type of combinatorial design that were developed in the 1940s in the design of statistical experiments. In 1947, Rao proved a lower bound on the size of any orthogonal array, and raised the problem of constructing arrays of minimum size. Kuperberg, Lovett and Peled (2017) gave a nonconstructive existence proof of orthogonal arrays whose size is nearoptimal (i.e., within a polynomial of Rao's lower bound), leaving open the question of an algorithmic construction. We give the first explicit, deterministic, algorithmic construction of orthogonal arrays achieving nearoptimal size for all parameters. Our construction uses algebraic geometry codes. In pseudorandomness, the notions of $t$independent generators or $t$independent hash functions are equivalent to orthogonal arrays. Classical constructions of $t$independent hash functions are known when the size of the codomain is a prime power, but very few constructions are known for an arbitrary codomain. Our construction yields algorithmically efficient $t$independent hash functions for arbitrary domain and codomain.
 Publication:

arXiv eprints
 Pub Date:
 May 2024
 DOI:
 10.48550/arXiv.2405.08787
 arXiv:
 arXiv:2405.08787
 Bibcode:
 2024arXiv240508787H
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Computational Complexity;
 Mathematics  Combinatorics;
 Mathematics  Statistics Theory
 EPrint:
 doi:10.1145/3618260.3649642