Singer 8-arcs of Mathon type in PG(2,2^7)

TL;DR

This paper introduces a special class of Mathon maximal arcs of degree 8 in PG(2,2^7) that admit a Singer group, describes their structure, and extends these arcs to infinite families in larger projective planes.

Contribution

It identifies and characterizes a new class of Mathon maximal arcs with Singer groups in PG(2,2^7) and extends these to infinite families in PG(2,2^k) for odd k divisible by 7.

Findings

Existence of a special class of Mathon arcs with Singer groups in PG(2,2^7)

Explicit description of these arcs and their properties

Extension of these arcs to infinite families in larger projective planes

Abstract

In a former paper the authors counted the number of non-isomorphic Mathon maximal arcs of degree 8 in PG(2,2^h), h not 7 and prime. In this paper we will show that in PG(2,2^7) 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(2,2^7) 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.