Sharp groups, two-weight codes and maximal arcs. (Q2462325)

Let \(G\) be a permutation group acting on a finite set \(\Omega\) of size \(n\) and let \(A\subseteq\{0,1,\dots,n-1\}\). \(G\) is said to be `\((A,n)\)-sharp' iff \(|G|=\prod_{a\in A}(n-a)\) holds. \(G\) is called `geometric' iff the pointwise stabilizer in \(G\) of any subset of \(\Omega\) acts transitively on the set of points that it does not fix, if there are any. \(G\) is said to have `type' \((A,n)\) iff any \(1\neq g\in G\) has exactly \(a\) fixed-points for some \(a\in A\). Note that any geometric group of type \((A,n)\) is \((A,n)\)-sharp. In the present paper the case \(A=\{0,a\}\) for some \(a\geq 2\) is studied in some detail. The primitive \((\{0,a\},n)\)-sharp groups have been almost completely classified by \textit{D. P. Brozovic} [Commun. Algebra 24, No. 12, 3979-3994 (1996; Zbl 0859.20003), ibid. 28, No. 4, 2103-2129 (2000; Zbl 0966.20002)]. Here the imprimitive case is studied by considering a different kind of geometrical structure: It is supposed that \(G\) leaves invariant a system of blocks of size \(q\) where \(q\) is a power of a prime \(p\) such that the subgroup stabilizing each of these blocks setwise contains an elementary Abelian \(p\)-group which is normal in the whole group, acting regularly in its induced action on each block. In this situation the author is able to construct a linear code over \(\text{GF}(q)\) whose non-trivial code words have only two different Hamming weights. Using the correspondence of \textit{R. Calderbank} and \textit{W. M. Kantor} [Bull. Lond. Math. Soc. 18, 97-122 (1986; Zbl 0582.94019)] between two-weight linear codes and certain particular subsets of finite projective geometries the two-weight codes arising in this way can be described (Theorem 3). Considering hyperplane complements in \(\text{PG}(k-1,q)\) allows to determine to some extent several cases of subgroups \(G\leq\text{AGL}(k,q)\) which are \((\{0,q^{k-1}\},q^k)\)-sharp (Theorem 4).