On the (29, 5)-Arcs in PG(2, 7) and Some Generalized Arcs in PG(2, q)

Abstract

Using an exhaustive computer search, we prove that the number of inequivalent (29,5)-arcs in PG(2,7) is exactly 22. This generalizes a result of Barlotti (see Barlotti, A. Some Topics in Finite Geometrical Structures, 1965), who constructed the first such arc from a conic. Our classification result is based on the fact that arcs and linear codes are related, which enables us to apply an algorithm for classifying the associated linear codes instead. Related to this result, several infinite families of arcs and multiple blocking sets are constructed. Lastly, the relationship between these arcs and the Barlotti arc is explored using a construction that we call transitioning.