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 non-constructive existence proof of orthogonal arrays whose size is near-optimal (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 near-optimal size for all parameters. Our construction uses algebraic geometry codes. In pseudorandomness, the notions of -independent generators or -independent hash functions are equivalent to orthogonal arrays. Classical constructions of -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 -independent hash functions for arbitrary domain and codomain.
Cite
@article{arxiv.2405.08787,
title = {Explicit Orthogonal Arrays and Universal Hashing with Arbitrary Parameters},
author = {Nicholas Harvey and Arvin Sahami},
journal= {arXiv preprint arXiv:2405.08787},
year = {2024}
}