The geometry of intersecting codes and applications to additive combinatorics and factorization theory

TL;DR

This paper explores the geometric structure of intersecting codes, improves bounds on their length, and connects these codes to additive combinatorics and algebraic factorizations, providing new theoretical insights and constructions.

Contribution

It establishes a geometric characterization of intersecting codes, improves bounds on their length, and links coding theory to the Davenport constant and algebraic factorizations.

Findings

Nondegenerate intersecting codes correspond to point sets not contained in two hyperplanes.

New explicit constructions of short intersecting codes.

Asymptotic bounds on the weighted 2-wise Davenport constant.

Abstract

Intersecting codes are linear codes where every two nonzero codewords have non-trivially intersecting support. In this article we expand on the theory of this family of codes, by showing that nondegenerate intersecting codes correspond to sets of points (with multiplicites) in a projective space that are not contained in two hyperplanes. This correspondence allows the use of geometric arguments to demonstrate properties and provide constructions of intersecting codes. We improve on existing bounds on their length and provide explicit constructions of short intersecting codes. Finally, generalizing a link between coding theory and the theory of the Davenport constant (a combinatorial invariant of finite abelian groups), we provide new asymptotic bounds on the weighted $2$-wise Davenport constant. These bounds then yield results on factorizations in rings of algebraic integers and related structures.