Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Explicit Orthogonal Arrays and Universal Hashing with Arbitrary Parameters

Nicholas Harvey, Arvin Sahami

TL;DR

The paper provides explicit, deterministic constructions of near-optimal orthogonal arrays for arbitrary parameters by leveraging algebraic geometry codes, achieving sizes $s$ close to Rao’s lower bound; Reed-Solomon instantiations offer practical bounds, while AG-codes yield near-optimal performance across all regimes with polynomial-time construction. It also establishes a tight connection between orthogonal arrays and $t$-independent hash families, delivering efficient representations, evaluation, and construction times (GRH-based improvements). The work unifies coding-theory techniques with pseudorandomness objectives, delivering concrete OA-based hash families that work over general alphabets and offering randomized alternatives with near-optimal sizes. These results advance deterministic, scalable combinatorial designs and hashing schemes applicable to a broad range of pseudorandomness and algorithmic tasks.

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 $t$-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.

Explicit Orthogonal Arrays and Universal Hashing with Arbitrary Parameters

TL;DR

The paper provides explicit, deterministic constructions of near-optimal orthogonal arrays for arbitrary parameters by leveraging algebraic geometry codes, achieving sizes close to Rao’s lower bound; Reed-Solomon instantiations offer practical bounds, while AG-codes yield near-optimal performance across all regimes with polynomial-time construction. It also establishes a tight connection between orthogonal arrays and -independent hash families, delivering efficient representations, evaluation, and construction times (GRH-based improvements). The work unifies coding-theory techniques with pseudorandomness objectives, delivering concrete OA-based hash families that work over general alphabets and offering randomized alternatives with near-optimal sizes. These results advance deterministic, scalable combinatorial designs and hashing schemes applicable to a broad range of pseudorandomness and algorithmic tasks.

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.
Paper Structure (15 sections, 17 theorems, 38 equations, 2 algorithms)

This paper contains 15 sections, 17 theorems, 38 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Previous constructions.
  3. Our results.
  4. Hash Functions
  5. Applications
  6. The abstract construction using codes
  7. Construction using Reed-Solomon codes
  8. The prime.
  9. The codes.
  10. Construction using algebraic geometry codes
  11. Algebraic geometry codes.
  12. Using AG codes for orthogonal arrays.
  13. Implementing a $t$-independent hash function
  14. Construction using random linear codes
  15. Open Questions

Key Result

Theorem 1

For any $m,n,t$ (with $n$ not necessarily a prime power), there is an explicit description of an orthogonal array with parameters $[s,m,n,t]$ where $s=(cmn/t)^{35t}$, and $c$ is a universal constant. The array can be constructed by a deterministic algorithm with runtime $\operatorname{poly}(sm)$.

Theorems & Definitions (27)

  • Theorem 1: Informal
  • Claim 2
  • Theorem 3: Informal
  • Lemma 4
  • proof : Proof (of Lemma \ref{['lem:BuildOAWorks']})
  • Corollary 5
  • proof
  • Theorem 6
  • Theorem 7: Linnik's theorem
  • Corollary 8
  • ...and 17 more