On small dense arcs in Galois planes of square order
A \((k,n)\)-arc \({\mathcal K}\) in the projective plane \({\mathcal P}\) over a field \({\mathbb F}_q\) with \(q=p^{2r}\) elements is a set of \(k\) points intersecting each line in at most \(n\) points; \({\mathcal K}\) is said to be dense if each point of \({\mathcal P}\) is on a line containing \(2\) points of \({\mathcal K}\). Maximal or complete \((k,n)\)-arcs are of interest in coding theory; \((k,2)\)-arcs are known as \(k\)-arcs. Obviously, a \(k\)-arc is complete if, and only if, it is dense. In this nicely written paper, the authors show the existence of small dense \((k,4)\)-arcs such that \({\mathcal K}\) is contained in the union of two conics or, for even \(q\), of two hyperovals. If \(p=2\) and \(\ell \geq 2\log_2q\) then there exists a dense \((k,4)\)-arc with \(k \leq 2\ell(p^r+2)\); for odd \(q\), the bound is slightly larger. Moreover, for \(p=2\), the authors construct \(4(2^r{-}1)\)-arcs, which are complete for small values of \(r\).
scientific article; zbMATH DE number 1934668