Abstract
For q odd and n > 1 odd, a new infinite family of large complete arcs K′ in PG(2, q n) is constructed from complete arcs K in PG(2, q) which have the following property with respect to an irreducible conic \({\mathcal{C}}\) in PG(2, q): all the points of K not in \({\mathcal{C}}\) are all internal or all external points to \({\mathcal{C}}\) according as q ≡ 1 (mod 4) or q ≡ 3 (mod 4).
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Abatangelo V., Fisher J.C., Korchmáros G., Larato B.: On the mutual position of two irreducible conics in PG(2, q), q odd, submitted.
Boros E., Szőnyi T.: On the sharpness of a theorem of B. Segre. Combinatorica 6, 261–268 (1986)
Cossidente A.: A new proof of the existence of (q 2 − q + 1)-arcs in PG(2, q 2. J. Geom. 53, 37–40 (1995); correction 59, 32–33 (1997).
Cossidente A., Korchmáros G.: The Hermitian function field arising from a cyclic arc in a Galois plane. Geometry, Combinatorial Designs and Related Structures, London Mathematical Society Lecture Note Series, vol. 245, pp. 63–68. Cambridge University Press, Cambridge (1997).
Cossidente A., Korchmáros G.: The algebraic envelope associated to a complete arc. Rend. Circ. Mat. Palermo Suppl. 51, 9–24 (1998)
Ebert G.: Partitioning projective geometries into caps. Can. J. Math. 37, 1163–1175 (1985)
Fisher J.C., Hirschfeld J.W.P., Thas J.A.: Complete arcs in planes of square order. Combinatorics ’84. North-Holland Mathematics Studies, vol. 123, pp. 243–250. North-Holland, Amsterdam (1986).
Giulietti M.: On plane arcs contained in cubic curves. Finite Fields Appl. 8, 69–90 (2002)
Hirschfeld J.W.P., Korchmáros G.: Embedding an arc into a conic in a finite plane. Finite Fields Appl. 2, 274–292 (1996)
Hirschfeld J.W.P., Korchmáros G.: On the number of rational points on an algebraic curve over a finite field. Bull. Belg. Math. Soc. Simon Stevin. 5, 313–340 (1998)
Hirschfeld J.W.P., Korchmáros G., Torres F.: Algebraic Curves Over a Finite Field, pp. xx+696. Princeton University Press, Princeton and Oxford (2008).
Kestenband B.: Unital intersections in finite projective planes. Geom. Dedicata 11, 107–117 (1981)
Korchmáros G.: A combinatorial characterisation of the dihedral subgroups of order 2(p r + 1) of PGL (2, p r). Geom. Dedicata 9, 381–384 (1980)
Korchmáros G., Sonnino A.: On arcs sharing the maximum number of points with ovals in cyclic affine planes of odd order. J. Combin. Designs (2009). doi:10.1002/jcd.20220.
Valentini R.C., Madan M.L.: A Hauptsatz of L.E. Dickson and Artin–Schreier extensions. J. Reine Angew. Math. 318, 156–177 (1980)
Voloch J.F.: On the completeness of certain plane arcs. Eur. J. Combin. 8, 453–456 (1987)
Voloch J.F.: On the completeness of certain plane arcs II. Eur. J. Combin. 11, 491–496 (1990)
Voloch J.F.: Arcs in projective planes over prime fields. J. Geom. 38, 198–200 (1990)
Voloch J.F. et al.: Complete arcs in Galois planes of non-square order. In: Hirschfeld, J.W.P. (eds) Advances in Finite Geometries and Designs, Isle of Thorns 1990., pp. 401–405. Oxford University Press, Oxford (1991)
Waterhouse W.C.: Abelian varieties over finite fields. Ann. Sci. École Norm. Sup. 2, 521–560 (1969)
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Communicated by Ron Mullin, Rainer Steinwandt.
Dedicated to Spyros Magliveras on the occasion of his 70th birthday.
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Korchmáros, G., Pace, N. Infinite family of large complete arcs in PG(2, q n), with q odd and n > 1 odd. Des. Codes Cryptogr. 55, 285–296 (2010). https://doi.org/10.1007/s10623-009-9343-6
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DOI: https://doi.org/10.1007/s10623-009-9343-6