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

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.