An infinite family of 7-designs (Q5948965)

The author studies designs invariant under the affine group \(G = E \cdot L\) over the field with two elements in its 3-transitive action on the \(2^n\) points of the underlying vector space \(V = F_2^n\). Here \(E\) is an elementary abelian group of order \(2^n\) and \(L\) is the \(n\)-dimensional general linear group \(\text{GL}_n(2)\). \textit{W. O. Alltop} [5-designs in affine spaces, Pac. J. Math. 39, 547-551 (1971; Zbl 0239.05012)] was the first to use this group to construct \(t\)-designs with \(t > 3\). As \(G\) is 3-transitive, for every cardinality \(k\), every orbit of \(G\) on \(k\)-subsets of the point set is a 3-design. Alltop used single orbits, proving that such an orbit is a 5-design if and only if it is a 4-design and producing an explicit example of a 5-design with \(k = 24\) in case \(n = 8\). Here the author studies the situation in more detail in order to construct infinite families of non-simple \(t\)-designs whose blocks are unions of \(G\). He restricts his attention to block sizes at most \(8\). The main result is a family 7-\((2^n,8,45)\), \(n \geq 6\), of non-simple designs. He also obtains designs 5-\((2^n,6,3)\) for every \(n \geq 3\) and 5-\((2^n,7,7(2^n-16)/2)\) for every even \(n \geq 6\).