We show that any affine block design D = (P, B) is a subset of a suitable commutative group G_D, with the property that a k-subset of P is a block of D if and only if its k elements sum up to zero. As a consequence, the group of automorphisms of any affine design D is the group of automorphisms of G_D that leave P invariant. Whenever k is a prime p, G_D is an elementary abelian p-group.
Caggegi, A., Falcone, G., Pavone, M. (2021). Additivity of affine designs. JOURNAL OF ALGEBRAIC COMBINATORICS, 53(3), 755-770 [10.1007/s10801-020-00941-8].
Additivity of affine designs
Caggegi, Andrea;Falcone, Giovanni
;Pavone, Marco
2021-05-01
Abstract
We show that any affine block design D = (P, B) is a subset of a suitable commutative group G_D, with the property that a k-subset of P is a block of D if and only if its k elements sum up to zero. As a consequence, the group of automorphisms of any affine design D is the group of automorphisms of G_D that leave P invariant. Whenever k is a prime p, G_D is an elementary abelian p-group.
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