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

Martino Borello, Wolfgang Schmid, Martin Scotti

TL;DR

The paper develops a geometric and combinatorial framework for intersecting codes, showing they correspond to non-2-cohyperplanar point sets in projective space and linking minimal/outer-minimal codes to strong blocking sets and the avoidance property. It then derives sharp bounds and explicit constructions for the length of the shortest intersecting codes, including AG-code-based and expander-graph-based methods, yielding asymptotically good families and concrete small-dimension examples. A central contribution is the connection between intersecting codes and generalized Davenport constants, including a broad W-weighted framework that encompasses elementary abelian groups and Galois actions, with both asymptotic and explicit results. Finally, the authors apply these coding-theoretic insights to factorization theory in algebraic number fields, showing how Davenport-type invariants govern irreducible factorizations in rings of integers and suggesting new avenues for interactions between coding theory, additive combinatorics, and algebraic number theory.

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.

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

TL;DR

The paper develops a geometric and combinatorial framework for intersecting codes, showing they correspond to non-2-cohyperplanar point sets in projective space and linking minimal/outer-minimal codes to strong blocking sets and the avoidance property. It then derives sharp bounds and explicit constructions for the length of the shortest intersecting codes, including AG-code-based and expander-graph-based methods, yielding asymptotically good families and concrete small-dimension examples. A central contribution is the connection between intersecting codes and generalized Davenport constants, including a broad W-weighted framework that encompasses elementary abelian groups and Galois actions, with both asymptotic and explicit results. Finally, the authors apply these coding-theoretic insights to factorization theory in algebraic number fields, showing how Davenport-type invariants govern irreducible factorizations in rings of integers and suggesting new avenues for interactions between coding theory, additive combinatorics, and algebraic number theory.

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 -wise Davenport constant. These bounds then yield results on factorizations in rings of algebraic integers and related structures.
Paper Structure (18 sections, 30 theorems, 109 equations, 6 tables)

This paper contains 18 sections, 30 theorems, 109 equations, 6 tables.

Table of Contents

  1. Intersecting codes: definitions and fundamental properties
  2. The geometry of intersecting codes
  3. On the size of small non-2-cohyperplanar sets
  4. Lower bounds
  5. Asymptotic lower bounds
  6. Upper bounds
  7. Explicit examples for low dimensions and small base fields
  8. Small explicit constructions for large dimensions
  9. Algebraic geometry intersecting codes
  10. A construction using expander graphs
  11. On the 2-wise weighted Davenport constants
  12. The general setting
  13. Our generalization
  14. Asymptotic bounds
  15. Applications to factorization theory
  16. ...and 3 more sections

Key Result

Lemma 1.5

Let $\mathcal{C}$ be an intersecting $[N, K, D]_{q^k}$ code and $\mathcal{I}$ an intersecting $[n, k, d]_q$ code. Then $\mathcal{I} \square \mathcal{C}$ is an intersecting $[Nn, Kk, \geq Dd]_q$ code.

Theorems & Definitions (111)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Lemma 1.5
  • Definition 1.6
  • Lemma 1.7
  • proof
  • Definition 1.8
  • Proposition 1.9
  • ...and 101 more