Maximum scattered linear sets and complete caps in Galois spaces

TL;DR

This paper constructs infinite families of maximal rank scattered linear sets in projective spaces over finite fields and explores their applications to complete caps in affine geometries, addressing longstanding size bounds.

Contribution

It provides explicit constructions of maximal rank scattered linear sets and their application to creating large complete caps in affine spaces, solving a key open problem.

Findings

Constructed infinite families of scattered linear sets of maximal rank.

Derived complete caps in affine spaces with sizes close to theoretical lower bounds.

Addressed the open problem of minimal size of complete caps in even-dimensional affine spaces.

Abstract

Explicit constructions of infinite families of scattered ${\mathbb F}_q$--linear sets in $PG(r-1,q^t)$ of maximal rank $\frac{rt}2$, for $t$ even, are provided. When $q=2$ and $r$ is odd, these linear sets correspond to complete caps in $AG(r,2^t)$ fixed by a translation group of size $2^{\frac{rt}2}$. The doubling construction applied to such caps gives complete caps in $AG(r+1,2^t)$ of size $2^{\frac{rt}2+1}$. For Galois spaces of even dimension greater than $2$ and even square order, this solves the long-standing problem of establishing whether the theoretical lower bound for the size of a complete cap is substantially sharp.