Optimizing Extension Techniques for Discovering Non-Algebraic Matroids
Michael Bamiloshin, Oriol Farràs
TL;DR
This work refines combinatorial and information-theoretic tools for distinguishing non-algebraic matroids by optimizing Dress-Lovász and Ahlswede-Körner extension techniques. It reduces AK/DL computations through flats-based checks and LP formulations, enabling discovery of new non-algebraic matroids on 9–10 points and yielding improved lower bounds for secret-sharing ports. The results advance matroid classification beyond Frobenius-flock methods, expanding the landscape of non-algebraic and non-almost-entropic matroids with concrete computational methods. Practically, the enhanced AK-based LP approach yields tighter information-ratio bounds for secret sharing, with implications for constructing and evaluating information-theoretic access structures. The work also provides open-source tooling to reproduce and extend these findings.
Abstract
In this work, we revisit some combinatorial and information-theoretic extension techniques for detecting non-algebraic matroids. These are the Dress-Lovász and Ahlswede-Körner 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-Körner extension property to find better lower bounds on the information ratio of secret sharing schemes for ports of non-algebraic matroids.