(2,1)-separating systems beyond the probabilistic bound

TL;DR

This paper introduces new algebraic geometry-based lower bounds for intersecting codes over large alphabets and constructs binary (2,1)-separating systems surpassing probabilistic bounds, resolving a 30-year open question.

Contribution

It provides the first explicit construction of (2,1)-separating systems exceeding probabilistic bounds, using algebraic geometry techniques.

Findings

New lower bounds for intersecting codes over large alphabets.

Construction of binary (2,1)-separating systems with higher asymptotic rate.

Answer to the open question on the exactness of probabilistic bounds for (2,1)-separating systems.

Abstract

Building on previous results of Xing, we give new lower bounds on the rate of intersecting codes over large alphabets. The proof is constructive, and uses algebraic geometry, although nothing beyond the basic theory of linear systems on curves. Then, using these new bounds within a concatenation argument, we construct binary (2,1)-separating systems of asymptotic rate exceeding the one given by the probabilistic method, which was the best lower bound available up to now. This answers (negatively) the question of whether this probabilistic bound was exact, which has remained open for more than 30 years. (By the way, we also give a formulation of the separation property in terms of metric convexity, which may be an inspirational source for new research problems.)