  We consider the extremal family of graphs of order $2^n$ in which no two
vertices have identical neighbourhoods, yet the adjacency matrix has rank only
$n$ over the field of two elements. A previous result from algebraic geometry
shows that such graphs exist for all even $n$ and do not exist for odd $n$. In
this paper we provide a new combinatorial proof for this result, offering
greater insight to the structure of graphs with these properties. We introduce
a new graph product closely related to the Kronecker product, followed by a
construction for such graphs for any even $n$. Moreover, we show that this is
an infinite family of strongly-regular quasi-random graphs whose signed
adjacency matrices are symmetric Hadamard matrices. Conversely, we provide a
combinatorial proof that for all odd $n$, no twin-free graphs of minimal
$\mathbb{F}_2$-rank exist, and that the next best-possible rank $(n+1)$ is
attainable, which is tight.
