A 3-cycle construction of complete arcs sharing $(q+3)/2$ points with a conic

Daniele Bartoli; Alexander A. Davydov; Stefano Marcugini; Fernanda Pambianco

doi:10.3934/amc.2013.7.319

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2013, Volume 7, Issue 3: 319-334. Doi: 10.3934/amc.2013.7.319

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A 3-cycle construction of complete arcs sharing $(q+3)/2$ points with a conic

1.
Department of Mathematics and Informatics, Perugia University, Perugia, 06123
2.
Institute for Information Transmission Problems (Kharkevich institute), Russian Academy of Sciences, GSP-4, Moscow, 127994

Received: July 31, 2012

Revised: May 31, 2013

Published: July 2013

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Abstract

In the projective plane $PG(2,q),$ $q\equiv 2$ $(\bmod~3)$ odd prime power, $ q\geq 11,$ an explicit construction of $\frac{1}{2}(q+7)$-arcs sharing $ \frac{1}{2}(q+3)$ points with an irreducible conic is considered. The construction is based on 3-orbits of some projectivity, called 3-cycles. For every $q,$ variants of the construction give non-equivalent arcs. It allows us to obtain complete $\frac{1}{ 2}(q+7)$-arcs for $q\leq 4523.$ Moreover, for $q=17,59$ there exist variants that are incomplete arcs. Completing these variants we obtained complete $( \frac{1}{2}(q+3)+\delta)$-arcs with $ \delta =4,$ $q=17,$ and $\delta =3,$ $q=59$; a description of them as union of some symmetrical objects is given.

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Mathematics Subject Classification: Primary: 51E21, 51E22; Secondary: 94B05.

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