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.

翻译：正交阵列是一种组合设计，于20世纪40年代在统计实验设计中发展而来。1947年，Rao证明了任何正交阵列的规模下界，并提出了构造最小规模阵列的问题。Kuperberg、Lovett和Peled（2017）给出了正交阵列存在的非构造性证明，其规模接近最优（即与Rao下界相差多项式因子），但算法构造问题仍未解决。我们首次给出了显式、确定性的算法构造，对任意参数均能实现接近最优规模的正交阵列。该构造利用了代数几何码。在伪随机性领域，t-独立生成器或t-独立哈希函数的概念与正交阵列等价。当值域大小为素数幂时，经典的t-独立哈希函数构造已知，但针对任意值域的构造极少。我们的构造为任意定义域和值域提供了算法高效的t-独立哈希函数。