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.
Highlights
Let Fq be the finite field with q elements, q a prime power
An (n, r )-arc K in PG(2, q) is an n-set in PG(2, q) such that each line contains at most r points of K and some lines contain exactly r points of K
We show how to construct these arcs from the well-known arc found by Barlotti [10] by exchanging some points, an operation called transition
Summary
Let Fq be the finite field with q elements, q a prime power. We denote by PG(2, q) the projective plane over Fq. For i = 1, 2, let Gi be a generator matrix of a projective [n, k, d]q code Ci and let Ki be the n-set in PG(k − 1, q) corresponding to the n columns of Gi . (n, r )-arcs and ( f , m)-blocking sets are equivalent objects.
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