Flag-transitive 2-designs arising from line-spreads in PG\((2n-1,2)\) (Q1306496)

The number of lines in \(\text{PG}(2n-1,2)\) whose orbit under the subgroup of index \(3\) in the Singer group is a spread is given by \((2^{2n}-1)(2^n+(-1)^{n+1})^2/27\). Such lines give rise (by a developement) to a flag-transitive \(2\)-\((2^{2n},4,1)\) design \(D\). Cases in which \(D\) is a \(\text{AG}(n,4)\) are determined.