Explicit Orthogonal Arrays And Universal Hashing With Arbitrary Parameters
Harvey Nicholas, Sahami Arvin. Arxiv 2024
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.
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