On sizes of complete arcs in PG(2,q)

New upper bounds on the smallest size t_{2}(2,q) of a complete arc in the projective plane PG(2,q) are obtained for 853 <= q <= 4561 and q\in T1\cup T2 where T1={173,181,193,229,243,257,271,277,293,343,373,409,443,449,457, 461,463,467,479,487,491,499,529,563,569,571,577,587,593,599,601,607,613,617,619,631, 641,661,673,677,683,691, 709}, T2={4597,4703,4723,4733,4789,4799,4813,4831,5003,5347,5641,5843,6011,8192}. From these new bounds it follows that for q <= 2593 and q=2693,2753, the relation t_{2}(2,q) < 4.5\sqrt{q} holds. Also, for q <= 4561 we have t_{2}(2,q) < 4.75\sqrt{q}. It is showed that for 23 <= q <= 4561 and q\in T2\cup {2^{14},2^{15},2^{18}}, the inequality t_{2}(2,q) < \sqrt{q}ln^{0.75}q is true. Moreover, the results obtained allow us to conjecture that this estimate holds for all q >= 23. The new upper bounds are obtained by finding new small complete arcs with the help of a computer search using randomized greedy algorithms. Also new constructions of complete arcs are proposed. These constructions form families of k-arcs in PG(2,q) containing arcs of all sizes k in a region k_{min} <= k <= k_{max} where k_{min} is of order q/3 or q/4 while k_{max} has order q/2. The completeness of the arcs obtained by the new constructions is proved for q <= 1367 and 2003 <= q <= 2063. There is reason to suppose that the arcs are complete for all q > 1367. New sizes of complete arcs in PG(2,q) are presented for 169 <= q <= 349 and q=1013,2003.Comment: 27 pages, 4 figures, 5 table

On the size of complete caps in PG(3,2h)

AbstractLet m2′(3,q) be the largest value of k (k<q2+1) for which there exists a complete k-cap in PG(3,q), q even. In this paper, the known upper bound on m2′(3,q) is improved. We also improve a number of intervals, for k, for which there does not exist a complete k-cap in PG(3,q), q even

The small weight codewords of the functional codes associated to non-singular hermitian varieties

This article studies the small weight codewords of the functional code C (Herm) (X), with X a non-singular Hermitian variety of PG(N, q (2)). The main result of this article is that the small weight codewords correspond to the intersections of X with the singular Hermitian varieties of PG(N, q (2)) consisting of q + 1 hyperplanes through a common (N - 2)-dimensional space I , forming a Baer subline in the quotient space of I . The number of codewords having these small weights is also calculated. In this way, similar results are obtained to the functional codes C (2)(Q), Q a non-singular quadric (Edoukou et al., J. Pure Appl. Algebra 214:1729-1739, 2010), and C (2)(X), X a non-singular Hermitian variety (Hallez and Storme, Finite Fields Appl. 16:27-35, 2010)

On Saturating Sets in Small Projective Geometries

AbstractA set of points, S⊆PG(r, q), is said to be ϱ -saturating if, for any point x∈PG(r, q), there exist ϱ+ 1 points in S that generate a subspace in which x lies. The cardinality of a smallest possible set S with this property is denoted by k(r, q,ϱ ). We give a short survey of what is known about k(r, q, 1) and present new results for k(r, q, 2) for small values of r and q. One construction presented proves that k(5, q, 2) ≤ 3 q+ 1 forq= 2, q≥ 4. We further give an upper bound onk (ϱ+ 1, pm, ϱ)