We use the geometric description to determine the best parameters of quaternary additive codes of small length. Only one open question remains for length ≤ 13. Among our results are the non-existence of [12, 7, 5]-codes and [12, 4.5, 7]-codes as well as the existence of a [13, 7.5, 5]−code.
In the Galois projective plane of square order q, we show the existence of small dense (k; 4)-arcs whose points lie on two conics for q odd, and on two hyperovals for q even. We provide an explicit construction of (4 √ q − 4; 2)-arcs for q even, and we also show that they are complete as far as q 6 1024.
In the binary projective spaces PG(n, 2) k-caps are called large if k > 2 n−1 and small if k ≤ 2 n−1. In this paper we propose new constructions producing infinite families of small binary complete caps.
A trivial upper bound on the size k of an arc in an r-net is k ≤ r + 1. It has been known for about 20 years that if the r-net is Desarguesian and has odd order, then the case k = r + 1 cannot occur, and k ≥ r − 1 implies that the arc is contained in a conic. In this paper, we show that actually the same must hold provided that the difference r − k does not… (More)
The n-dimensional finite projective space, P G(n, q), admits a cyclic model, in which the set of points of P G(n, q) is identified with the elements of the group Z q n +q n−1 +···+q+1. It was proved by Hall (1974, Math. Centre Tracts, 57, 1–26) that in the cyclic model of P G(2, q), the additive inverse of a line is a conic. The following generalization of… (More)