How to Find New Characteristic-Dependent Linear Rank Inequalities using Secret Sharing

TL;DR

This paper introduces a method using secret sharing concepts to generate characteristic-dependent linear rank inequalities, aiding in establishing bounds on information ratios in secret sharing schemes.

Contribution

It presents a novel theorem that produces characteristic-dependent linear rank inequalities leveraging secret sharing ideas, advancing the understanding of bounds in secret sharing.

Findings

Provides a new theorem for characteristic-dependent inequalities

Enables lower bounds on information ratios in linear secret sharing

Enhances techniques for analyzing secret sharing schemes

Abstract

Determining information ratios of access structures is an important problem in secret sharing. Information inequalities and linear rank inequalities play an important role for proving bounds. Characteristic-dependent linear rank inequalities are rank inequalities which are true over vector spaces with specific field characteristic. In this paper, using ideas of secret sharing, we show a theorem that produces characteristic-dependent linear rank inequalities. These inequalities can be used for getting lower bounds on information ratios in linear secret sharing.