A new family of 2-designs over GF\((q)\) admitting SL\(_m(q^l)\)
This paper deals with the existence of \(t\)-designs, with \(t=2\), over a finite field. A \(t\)-design over a finite field \(\text{GF} (q)\), \(q\) being a prime power, denoted by \(t\)-\((v,k, \lambda;q)\), is a pair \((V,B)\), \(V\) being a \(V\)-dimensional vector space over \(\text{GF} (q)\) and \(B\) a collection of \(k\)-dimensional subspaces of \(V\), with the property that each \(t\)-dimensional subspace of \(V\) is contained in exactly \(\lambda\) members of \(B\) \((0< t\leq k\leq v)\). The \(t\)-design is called trivial if \(B\) is the set of all \(k\)-dimensional subspaces of \(V\). Starting from a 2-design with a smaller automorphism graph, the author constructs a 2-design with a larger automorphism graph. The main result of this paper can be stated as follows: Given a nontrivial 2-\((l,3, \frac{q^3(q^{l-5}-1)} {q-1};q)\) design for \(l\) satisfying \(l=5\bmod 6(q-1)\) which admits the action of a Singer cycle \(Z_l\) of \(\text{GL}_l (q)\), a \(2\)-\((ml,3, \frac{q^3(q^{l-5}-1)} {q-1};q)\) design \((m\geq 3)\) admitting the action of \(\text{SL}_m (q^l)\) is constructed. This construction when applied to Suzuki's designs gives a new family of 2-designs over \(\text{GF} (q)\) which admit the \(\text{SL}_m (q^l)\) action.
scientific article; zbMATH DE number 1129446