Characteristic-Dependent Linear Rank Inequalities via Complementary Vector Spaces

TL;DR

This paper introduces new characteristic-dependent linear rank inequalities using an alternative approach involving complementary vector spaces, with applications demonstrating how network solvability depends on field characteristics.

Contribution

It presents novel characteristic-dependent inequalities via a different technique from previous methods, and applies these to construct networks with solvability properties tied to field characteristics.

Findings

Existence of networks solvable only over fields with certain characteristics.

Linear capacity tends to zero for fields outside the specified characteristic set.

New inequalities expand understanding of rank conditions in vector spaces.

Abstract

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 produce new characteristic-dependent linear rank inequalities by an alternative technique to the usual Dougherty's inverse function method [9]. We take up some ideas of Blasiak [4], applied to certain complementary vector spaces, in order to produce them. Also, we present some applications to network coding. In particular, for each finite or co-finite set of primes $P$, we show that there exists a sequence of networks $\mathcal{N}\left(k\right)$ in which each member is linearly solvable over a field if and only if the characteristic of the field is in $P$, and the linear capacity, over fields whose characteristic is not in $P$, $\rightarrow0$ as $k\rightarrow\infty$.