Tactical decompositions of designs over finite fields

Abstract

An automorphism group of an incidence structure $$\mathcal {I}$$ induces a tactical decomposition on $$\mathcal {I}$$ . It is well known that tactical decompositions of $$t$$ -designs satisfy certain necessary conditions which can be expressed as equations in terms of the coefficients of tactical decomposition matrices. In this article we present results obtained for tactical decompositions of $$q$$ -analogs of $$t$$ -designs, more precisely, of $$2$$ - $$(v,k,\lambda _2;q)$$ designs. We show that coefficients of tactical decomposition matrices of a design over finite field satisfy an equation system analog to the one known for block designs. Furthermore, taking into consideration specific properties of designs over the binary field $$\mathbb {F}_2$$ , we obtain an additional system of inequations for these coefficients in that case.