Complete (q 2 + q + 8)/2-caps in the spaces PG(3, q), q ≡ 2 (mod 3) an odd prime, and a complete 20-cap in PG(3, 5)

Abstract

Complete (q 2 + q + 8)/2-caps in the spaces P G(3, q), q ≡ 2 (mod 3) an odd prime, and a complete 20-cap in P G(3, 5) Abstract An infinite family of complete (q 2 + q + 8)/2-caps is constructed in PG (3, q) where q is an odd prime ≡ 2 ( mod 3), q ≥ 11. This yields a new lower bound on the second largest size of complete caps. A variant of our construction also produces one of the two previously known complete 20-caps in PG(3, 5). The associated code weight distribution and other combinatorial properties of the new (q 2 + q + 8)/2-caps and the 20-cap in PG(3, 5) are investigated. The updated table of the known sizes of the complete caps in PG(3, q) is given. As a byproduct, we have found that the unique complete 14-arc in PG(2, 17) contains 10 points on a conic. Actually, this shows that an earlier general result dating back to the Seventies fails for q = 17.Keywords Complete caps · Projective spaces of the dimension three · Projective planes Mathematics Subject Classifications (2000) 51E21 · 51E22 · 94B05 IntroductionLet PG(n, q) be the projective space of dimension n over the Galois field F q and let F * q = F q \ {0}. A k-cap in PG(n, q) is a set of k points, no three of which are collinear. A k-cap in

On sizes of complete caps in projective spaces PG(n, q) and arcs in planes PG(2, q)

“…and K 1 \P ⊂ Q, see [26,Lemma 2]. The q points of the form (1, v, −v, 4v 2 ) and the point A ∞ lie on tangents to Q through P [26, Lemma 1], while the rest of points lies on bisecants of Q through P so that on every bisecant exactly one point of Q is included to K 1 , cf.…”

“…Points of the unique 14-arc K 17 (4) are given in [26]: The first ten points lie on the conic 3x 2 1 + x 2 2 = 2x 2 0 and the last four are placed outside it. In addition, the first twelve points are the arc K(2) of Construction S. The semicolons separate the points into orbits of the stabilizer group.…”

“…In [2,7,25,26,54,72,75] to obtain the results of (2)-(9) Segre's Construction A is used with distinct points P and distinct rules for the choice of one point of the quadric on every bisecant through P . [7]; for q ≡ 1 (mod 4), q = 5, 9, in [72]; for q = 2 h with odd h > 3 in [31]; for all even q > 8 in [1]; for q = 8 in [3]; for q = 9 in [43, Table 3.2]; for q = 4 in [76].…”

“…Table 3 gives the known sizes of complete caps in PG (3, q). We used the sizes of complete caps and bounds from the works [1][2][3][4][5][6][7]13], [23, Table 1], [25,26,30,31,[37][38][39][40][41][42][43][44]54,56,57,[66][67][68][69]72,73,[75][76][77][78], see also the references therein. We applied the relations (2)- (19) and Theorem 6.…”

“…The complete 41-cap in PG (3,16) is noted in [68, Table 3 The sizes of the known complete k-caps in PG (3, q). T q = (q 2 + q + 4)/2 q t 2 (3, q) Sizes k of the known m 2 (3, q) m 2 (3, q) References complete caps with [23,25,26,40,41,43,54,69,72,73,75] also (11). The complete 51-cap in PG(3, 17) is described in [23].…”