CharacteristicDependent Linear Rank Inequalities via Complementary Vector Spaces
Abstract
A characteristicdependent 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 characteristicdependent 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 cofinite 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 $k\rightarrow\infty$.
 Publication:

arXiv eprints
 Pub Date:
 March 2019
 arXiv:
 arXiv:1903.11587
 Bibcode:
 2019arXiv190311587P
 Keywords:

 Computer Science  Information Theory;
 68P30
 EPrint:
 20 pages