Small complete caps in Galois affine spaces
Massimo Giulietti
Università di Perugia Dipartimento di Matematica e Informatica 06123 Perugia Italy 06123 Perugia Italy
DOI: 10.1007/s10801-006-0029-0
Abstract
Some new families of caps in Galois affine spaces AG( N, q) of dimension N\equiv 0(mod 4) and odd order q are constructed. Such caps are proven to be complete by using some new ideas depending on the concept of a regular point with respect to a complete plane arc. As a corollary, an improvement on the currently known upper bounds on the size of the smallest complete caps in AG( N, q) is obtained.
Pages: 149–168
Keywords: keywords affine space; complete cap; complete arc
Full Text: PDF
References
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2. J. Bierbrauer, S. Marcugini, and F. Pambianco, “The smallest size of a complete cap in P G(3, 7),” Discrete Math., to appear.
3. G.D. Cohen, A.C. Lobstein, and N.J.A. Sloane, “Further results on the covering radius of codes,” IEEE Trans. Inform. Theory, 32(5) (1986), 680-694.
4. G.D. Cohen, I. Honkala, S. Litsyn, and A.C. Lobstein, Covering Codes, North-Holland, Amsterdam, 1997.
5. A.A. Davydov, S. Marcugini, and F. Pambianco, “On saturating sets in projective spaces,” J. Combin. Theory Ser. A 103 (2003), 1-15.
6. A.A. Davydov, S. Marcugini, and F. Pambianco, “Complete caps in projective spaces” P G(n, q), J. Geom. 80 (2004), 23-30.
7. A.A. Davydov, G. Faina, S. Marcugini and F. Pambianco, “Computer search in projective planes for the sizes of complete arcs,” J. Geom. 82 (2005), 50-62.
8. A.A. Davydov and P.R.J. “ \ddot Osterg\ring ard, Recursive constructions of complete caps,” J. Statist. Planning Infer. 95 (2001), 167-173.
9. E.M. Gabidulin, A.A. Davydov, and L.M. Tombak, “Linear codes with covering radius 2 and other new covering codes,” IEEE Trans. Inform. Theory 37 (1991), 219-224.
10. M. Giulietti, Small complete caps in P G(N , q), q even, submitted.
11. R.L. Graham and N.J.A. Sloane, “On the covering radius of codes,” IEEE Trans. Inform. Theory 39 (1993), 209-214.
12. J.W.P. Hirschfeld, Projective Geometries over Finite Fields, Clarendon Press, Oxford 1998.
13. J.W.P. Hirschfeld and L. Storme, “The packing problem in statistics, coding theory and finite projective spaces: update 2001,” in: Blokhuis, A. (ed.) et al., Finite geometries. Proceedings of the fourth Isle of Thorns conference, Brighton, UK, April
2000. Dordrecht: Kluwer Academic Publishers. Dev. Math. 3 (2001), 201-246.
14. J.J.E. Imber and D.L. Wehlau, “A family of small complete caps in P G(n, 2),” European J. Combin., 24 (2003), 613-615.
15. P.R.J “ \ddot Osterg\ring ard, Computer search for small complete caps,” J. Geom 69 (2000), 172-179.
16. F. Pambianco and L. Storme, “Small complete caps in spaces of even characteristic,” J. Combin. Theory Ser. A 75 (1996), 70-84.
17. B. Segre, “On complete caps and ovaloids in three-dimensional Galois spaces of characteristic two,” Acta Arith. 5 (1959), 315-332.
18. B. Segre, “Propriet`a elementari relative ai segmenti ed alle coniche sopra un campo qualsiasi ed una congettura di Seppo Ilkka per il caso dei campi di Galois,” Ann. Mat. Pura Appl. IV Ser. 96 (1973), 289-337.
19. H. Stichtenoth, Algebraic Function Fields and Codes, Springer Verlag, Berlin-Heidelberg-New York, 1993.