Flat-containing and shift-blocking sets in $F_2^r$

TL;DR

This paper investigates the minimal size of subsets in binary vector spaces that contain certain flats or block translates, providing bounds on their sizes for various parameters.

Contribution

It introduces new bounds on the sizes of flat-containing and shift-blocking sets in binary vector spaces, advancing understanding of their combinatorial properties.

Findings

Derived new lower bounds for flat-containing sets.

Established upper bounds for shift-blocking sets.

Connected combinatorial properties with geometric configurations.

Abstract

For non-negative integers $r\ge d$, how small can a subset $C\subset F_2^r$ be, given that for any $v\in F_2^r$ there is a $d$-flat passing through $v$ and contained in $C\cup\{v\}$? Equivalently, how large can a subset $B\subset F_2^r$ be, given that for any $v\in F_2^r$ there is a linear $d$-subspace not blocked non-trivially by the translate $B+v$? A number of lower and upper bounds are obtained.