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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.

A concept of locally optimal (LO) linear covering codes is introduced in accordance with the concept of minimal saturating sets in projective spaces over …nite …elds. An LO code is nonshortening in the sense that removing any column from a parity check matrix we increase the code covering radius. Constructions and in…nite families of LO codes are described.… (More)

We give a geometric interpretation of additive quantum stabilizer codes in terms of sets of lines in binary symplectic space. It is used to obtain synthetic geometric constructions and non-existence results. In particular several open problems are removed from Grassl's database [13].

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.

- Alexander A. Davydov, Giorgio Faina, Stefano Marcugini, Fernanda Pambianco, A. A. Davydov, G. Faina +2 others
- 2009

More than thirty new upper bounds on the smallest size t2(2, q) of a complete arc in the plane PG(2, q) are obtained for 169 ≤ q ≤ 839. New upper bounds on the smallest size t2(n, q) of the complete cap in the space PG(n, q) are given for n = 3 and 25 The bounds are obtained by computer search for new small complete arcs and caps. New upper bounds on the… (More)

On the spectrum of sizes of complete caps in projective spaces PG(n,q) of small dimension Abstract. For projective spaces P G(n, q) of small dimension, new sizes of complete caps including small these are obtained. The corresponding tables are given. A generalization of Segre's construction of complete caps in P G(3, 2 h) is described. In P G(2, q), for q =… (More)