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

Ferret, Sandy; Storme, Leo

doi:10.1016/j.ffa.2003.08.005

Cited by 3 publications

(5 citation statements)

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“…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.…”

Section: The Families Of Complete Caps In Pg(3 Q)mentioning

confidence: 99%

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

et al. 2009

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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 ≤ q ≤ 97, q odd; n = 4 and q = 7, 8, 11, 13, 17; n = 5 and q = 5, 7, 8, 9; n = 6 and q = 4, 8. The bounds are obtained by computer search for new small complete arcs and caps. New upper bounds on the largest size m2(n, q) of a complete cap in PG(n, q) are given for q = 4, n = 5, 6, and q = 3, n = 7, 8, 9. The new lower bound 534 ≤ m2(8, 3) is obtained by finding a complete 534-cap in PG(8, 3). Many new sizes of complete arcs and caps are obtained. The updated tables of upper bounds for t2(n, q), n ≥ 2, and of the spectrum of known sizes for complete caps are given. Interesting complete caps in PG(3, q) of large size are described. A proof of the construction of complete caps in PG(3, 2 h ) announced in previous papers is given; this is modified from a construction of Segre. In PG(2, q), for q = 17, δ = 4, and q = 19, 27, δ = 3, we give complete ( 1 2 (q + 3) + δ)-arcs other than conics that share 1 2 (q + 3) points with an irreducible conic. It is shown that they are unique up to collineation. In PG(2, q), q ≡ 2 (mod 3) odd, we propose new constructions of 1 2 (q + 7)-arcs and show that they are complete for q ≤ 3701. Mathematics Subject Classification (2000). 51E21, 51E22, 94B05.

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Section: The Families Of Complete Caps In Pg(3 Q)mentioning

confidence: 99%

“…At the same time, the lower bounds are of order T q = (q 2 +q +4)/2. For references on bounds; see [2,23,43,44,51,54], [57, Theorem 4.2, Table 4.2(ii)], [72,73,75,76,78] and (21)- (24). So, our knowledge on m 2 (3, q) seems to be insufficient.…”

Section: On the Spectrum Of Sizes Of Complete Caps In Pg(3 Q)mentioning

confidence: 99%

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

et al. 2009

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“…. , 2(q − 2)} of (14) consists of the only element 2(q − 2), whence v − d = 2(q − 2). It implies u = v−d 2 + 1 = (q − 2) + 1 = v + 1 that contradicts to the second relation of (16).…”

Section: Lemma 4 Condition 1 Implies the Point Pmentioning

confidence: 99%

“…The results t 2 (3, 7) = 17 [6] and m 2 (3, 16) ≤ 242 [14] are recent. In this paper we have obtained the following new sizes of the complete k -caps in PG (3, q): k = 90, 91, 92 for q = 13, k = 154 for q = 17, and k = 188, 190, 191 for q = 19.…”

Section: On the Spectrum Of Complete Caps Sizes In P G(3 Q)mentioning

confidence: 99%

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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)

Davydov

Marcugini

Pambianco

2008

Des. Codes Cryptogr.

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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

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Caps inPG(n,q)withqeven and
Cao
¹
,
Ou
²

2014
Discrete Mathematics

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

Cited by 3 publications

References 13 publications

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

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

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)

Caps inPG(n,q)withqeven and
Cao
¹
,
Ou
²

2014
Discrete Mathematics

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

Cited by 3 publications

References 13 publications

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

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

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)

Caps inPG(n,q)withqeven andCao1, Ou2 2014Discrete Mathematics

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Caps inPG(n,q)withqeven and
Cao
¹
,
Ou
²

2014
Discrete Mathematics