Generating binary spaces. (Q1873815)

This very interesting and well written paper solves completely the following problem: let \(r \geq \rho \geq 3\) integers and denote \textbf{F}\(^r_2\) the elementary 2-group of rank \(r\). Then the maximum possible size of a generating subset \(A\) of \textbf{F}\(^r_2,\) such that not all element of \textbf{F}\(^r_2\) are representable as a sum of fewer then \(\rho\) elements of \(A\) is \((\rho +1)2^{r-\rho}.\) The paper gives a full description of all such generating subsets \(A\) with big enough cardinality. It gives a broad algebraic background of the main result, furthermore it also interprets it as a coding theoretical theorem and draws several consequences of it.