论文信息 - Computer search in projective planes for the sizes of complete arcs

Computer search in projective planes for the sizes of complete arcs

Abstract.The spectrum of possible sizes k of complete k-arcs in finite projective planes PG(2, q) is investigated by computer search. Backtracking algorithms that try to construct complete arcs joining the orbits of some subgroup of collineation group PΓ L (3, q) and randomized greedy algorithms are applied. New upper bounds on the smallest size of a complete arc are given for q = 41, 43, 47, 49, 53, 59, 64, 71 ≤ q ≤ 809, q ≠ 529, 625, 729, and q = 821. New lower bounds on the second largest size of a complete arc are given for q = 31, 41, 43, 47, 53, 125. Also, many new sizes of complete arcs are obtained for 31 ≤ q ≤ 167.

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

[2] H. Niederreiter,et al. Finite Fields: Encyclopedia of Mathematics and Its Applications. , 1997 .

[3] Massimo Giulietti,et al. Small complete caps inPG(2,q), forq an odd square , 2000 .

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

[5] Tim Penttila,et al. Irregular hyperovals inPG(2, 64) , 1994 .

[6] Olga Polverino,et al. Small minimal blocking sets and complete k-arcs in PG(2, p3) , 1999, Discret. Math..

[7] Gérard D. Cohen,et al. Linear Codes with Covering Radius and Codimension , 2001 .

[8] Fernanda Pambianco,et al. On the spectrum of the valuesk for which a completek- cap in PG(n, q) exists , 1998 .

[9] Rudolf Lide,et al. Finite fields , 1983 .

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

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

[12] Fernanda Pambianco,et al. On some 10-arcs for deriving the minimum order for complete arcs in small projective planes , 1999, Discret. Math..