Explicit Orthogonal Arrays and Universal Hashing with Arbitrary Parameters
<|reference_start|>Explicit Orthogonal Arrays and Universal Hashing with Arbitrary Parameters: 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 $t$-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.<|reference_end|>
@article{harvey2024explicit,
    title={Explicit Orthogonal Arrays and Universal Hashing with Arbitrary
  Parameters},
    author={Nicholas Harvey, Arvin Sahami},
    journal={arXiv preprint arXiv:2405.08787},
    year={2024},
    doi={10.1145/3618260.3649642},
    archivePrefix={arXiv},
    eprint={2405.08787},
    primaryClass={cs.DS cs.CC math.CO math.ST stat.TH}
}