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An infinite family of 7-designs. (English) Zbl 0983.05014
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 $$\text{GL}_n(2)$$. 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 $$n \geq 6$$.

##### MSC:
 05B05 Combinatorial aspects of block designs
##### Keywords:
non-simple $$t$$-designs
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