Optimizing Extension Techniques for Discovering Non-Algebraic Matroids

TL;DR

This paper improves extension techniques for identifying non-algebraic matroids, reducing computational complexity and discovering new examples, while also enhancing bounds on secret sharing schemes.

Contribution

It optimizes combinatorial and information-theoretic extension methods, enabling the discovery of larger non-algebraic matroids and better bounds on secret sharing.

Findings

New non-algebraic matroids on 9 and 10 points

Reduced computational complexity of extension techniques

Improved lower bounds on secret sharing information ratio

Abstract

In this work, we revisit some combinatorial and information-theoretic extension techniques for detecting non-algebraic matroids. These are the Dress-Lov\'asz and Ahlswede-K\"orner extension properties. We provide optimizations of these techniques to reduce their computational complexity, finding new non-algebraic matroids on 9 and 10 points. In addition, we use the Ahlswede-K\"orner extension property to find better lower bounds on the information ratio of secret sharing schemes for ports of non-algebraic matroids.