On the representability of the bi-uniform matroid

TL;DR

This paper proves the existence of efficient methods to find representations for all bi-uniform matroids over finite fields, advancing understanding of their structure and implications for secret sharing.

Contribution

It introduces the first efficient construction methods for representing all bi-uniform matroids over finite fields, including smaller fields in many cases.

Findings

Efficient methods for representing all bi-uniform matroids are established.

Representations over smaller finite fields are achieved in many cases.

The results have implications for secret sharing schemes.

Abstract

Every bi-uniform matroid is representable over all sufficiently large fields. But it is not known exactly over which finite fields they are representable, and the existence of efficient methods to find a representation for every given bi-uniform matroid has not been proved. The interest of these problems is due to their implications to secret sharing. The existence of efficient methods to find representations for all bi-uniform matroids is proved here for the first time. The previously known efficient constructions apply only to a particular class of bi-uniform matroids, while the known general constructions were not proved to be efficient. In addition, our constructions provide in many cases representations over smaller finite fields.