Characteristic-Dependent Linear Rank Inequalities via Complementary 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 $k\rightarrow\infty$.