New inductive constructions of complete caps in PG(N, q), q even

Some new families of small complete caps in PG(N, q), q even, are described. By using inductive arguments, the problem of the construction of small complete caps in projective spaces of arbitrary dimensions is reduced to the same problem in the plane. The caps constructed in this article provide an improvement on the currently known upper bounds on the size of the smallest complete cap in PG(N, q), N≥4, for all q≥23. In particular, substantial improvements are obtained for infinite values of q square, including q=22Cm, C≥5, m≥3; for q=2Cm, C≥5, m≥9, with C, m odd; and for all q≤218. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 177–201, 2010

[1]  Tamás Szőnyi,et al.  Small complete arcs in Galois planes , 1985 .

[2]  Stefano Marcugini,et al.  Complete caps in projective spaces PG (n, q) , 2004 .

[3]  Van H. Vu,et al.  Small Complete Arcs in Projective Planes , 2003, Comb..

[4]  Ernst M. Gabidulin,et al.  Linear codes with covering radius 2 and other new covering codes , 1991, IEEE Trans. Inf. Theory.

[5]  Massimo Giulietti,et al.  Quasi-Perfect Linear Codes With Minimum Distance $4$ , 2007, IEEE Transactions on Information Theory.

[6]  Giorgio Faina,et al.  On small dense arcs in Galois planes of square order , 2003, Discret. Math..

[7]  Patric R. J. Osterg Aa Rd.  Computer search for small complete caps , 2000 .

[8]  Yves Edel.  Extensions of Generalized Product Caps , 2004, Des. Codes Cryptogr..

[9]  A. C. Mukhopadhyay.  Lower Bounds on mt(r, s) , 1978, J. Comb. Theory, Ser. A.

[10]  F. MacWilliams,et al.  The Theory of Error-Correcting Codes , 1977 .

[11]  Stefano Marcugini,et al.  Computer search in projective planes for the sizes of complete arcs , 2005 .

[12]  Massimo Giulietti,et al.  Small complete caps in PG(N, q), q even , 2007 .

[13]  A. A. Davydov,et al.  A computer search for small complete caps , 1998 .

[14]  Gábor Korchmáros,et al.  New Examples of Complete k-Arcs in PG(2, q) , 1983, Eur. J. Comb..

[15]  N. Johnson.  Sharply Transitive Sets , 2007 .

[16]  Stefano Marcugini,et al.  On sharply transitive sets in PG(2,q) , 2008 .

[17]  B. Segre.  On complete caps and ovaloids in three-dimensional Galois spaces of characteristic two , 1959 .

[18]  J. Hirschfeld.  Projective Geometries Over Finite Fields , 1980 .

[19]  Fernanda Pambianco,et al.  Small Complete Caps in Spaces of Even Characteristic , 1996, J. Comb. Theory, Ser. A.

[20]  Massimo Giulietti,et al.  Small complete caps in Galois affine spaces , 2007 .

[21]  J. Hirschfeld,et al.  The packing problem in statistics, coding theory and finite projective spaces : update 2001 , 2001 .

[22]  J. Hirschfeld,et al.  The packing problem in statistics, coding theory and finite projective spaces , 1998 .