How to Find New Characteristic-Dependent Linear Rank Inequalities using Binary Matrices as a Guide

Abstract

In Linear Algebra over finite fields, a characteristic-dependent linear rank inequality is a linear inequality that holds by ranks of subspaces of a vector space over a finite field of determined characteristic, and does not in general hold over other characteristics. In this paper, we show a method to produce these inequalities using binary matrices with suitable ranks over different fields. In particular, for each $n\ge 7$, we produce $2\lfloor \frac{n-1}{2}\rfloor -4$ characteristic-dependent linear rank inequalities over $n$ variables. Many of the inequalities obtained are new but some of them imply the inequalities presented in [1,9].

arXiv admin note: text overlap with arXiv:1903.11587

References in corpus (2)

• Characteristic-Dependent Linear Rank Inequalities with Applications to Network Coding
• Characteristic-Dependent Linear Rank Inequalities via Complementary Vector Spaces