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

Iliya Bouyukliev, Eun Ju Cheon, Tatsuya Maruta and Tsukasa Okazaki
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Iliya Bouyukliev: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Veliko Tarnovo, Bulgaria
Eun Ju Cheon: Department of Mathematics and RINS, Gyeongsang National University, Jinju 52828, Korea
Tatsuya Maruta: Department of Mathematical Sciences, Osaka Prefecture University, Sakai, Osaka 599-8531, Japan
Tsukasa Okazaki: Department of Mathematical Sciences, Osaka Prefecture University, Sakai, Osaka 599-8531, Japan

Mathematics, 2020, vol. 8, issue 3, 1-16

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.

Keywords: projective plane; arc; blocking set; linear code; Griesmer code (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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