Generalised Evasive Subspaces

TL;DR

This paper introduces the concept of evasive subspaces relative to collections of subspaces, providing bounds, existence results, and connections to rank-metric codes, advancing understanding in combinatorial geometry and coding theory.

Contribution

It generalizes existing notions of evasiveness, establishes bounds for evasive subspaces, and links these concepts to rank-metric codes and combinatorial geometries.

Findings

Derived upper bounds for evasive subspace dimensions.

Proved existence of evasive spaces using graph theory methods.

Connected evasive spaces to rank-metric code properties.

Abstract

We introduce and explore a new concept of evasive subspace with respect to a collection of subspaces sharing a common dimension, most notably partial spreads. We show that this concept generalises known notions of subspace scatteredness and evasiveness. We establish various upper bounds for the dimension of an evasive subspace with respect to arbitrary partial spreads, obtaining improvements for the Desarguesian ones. We also establish existence results for evasive spaces in a non-constructive way, using a graph theory approach. The upper and lower bounds we derive have a precise interpretation as bounds for the critical exponent of certain combinatorial geometries. Finally, we investigate connections between the notion of evasive space we introduce and the theory of rank-metric codes, obtaining new results on the covering radius and on the existence of minimal vector rank-metric codes.