Singer 8-arcs of Mathon type in PG(2, 27)

Abstract

De Clerck et al. (J Comb Theory, 2011) counted the number of non-isomorphic Mathon maximal arcs of degree-8 in PG(2, 2 h ), h ≠ 7 and prime. In this article we will show that in PG(2, 27) a special class of Mathon maximal arcs of degree-8 arises which admits a Singer group (i.e. a sharply transitive group) on the 7 conics of these arcs. We will give a detailed description of these arcs, and then count the total number of non-isomorphic Mathon maximal arcs of degree-8. Finally we show that the special arcs found in PG(2, 27) extend to two infinite families of Mathon arcs of degree-8 in PG(2, 2 k ), k odd and divisible by 7, while maintaining the nice property of admitting a Singer group.