The theorem of B. Segre mentioned in the title states that a complete arc of PG(2, q), q even which is not a hyperoval consists of at most q-√q+1 points. In the first part of our paper we prove this theorem to be sharp for q=s 2 by constructing complete q-√q+1-arcs. Our construction is based on the cyclic partition of PG(2, q) into disjoint Baer-subplanes. (See Bruck ). In his paper  Kestenband constructed a class of (q-√q+1)-arcs but he did not prove their completeness. In the second part of our paper we discuss the connections between Kestenband's and our constructions. We prove that these constructions result in isomorphic (q-√q+1)-arcs. The proof of this isomorphism is based on the existence of a traceorthogonal normal basis in GF(q 3 ) over GF(q), and on a representation of GF(q) 3 in GF(q 3 ) 3 indicated in Jamison .
All Science Journal Classification (ASJC) codes
- Discrete Mathematics and Combinatorics
- Computational Mathematics
- AMS subject classification (1980): 05B25, 51E15, 51E20