Computer search for small complete caps

Östergrd, Patric R. J.

doi:10.1007/bf01237484

Cited by 21 publications

(32 citation statements)

References 6 publications

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“…The smallest known size of a complete cap in PG (3,8) is 20 [66]. We obtained three inequivalent complete 20-caps in PG (3,8) with the stabilizer groups D 5 , Z 2 , S 3 of orders 10, 2, 6.…”

Section: Some Complete Caps In Pg(3 Q) With Boundary Sizesmentioning

confidence: 89%

“…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%

“…The complete k-caps with k = 17 in PG (3,7), k = 20 in PG (3,8), k = 24 in PG (3,9), and k = 30 in PG (3,11), are obtained in [66]. In [13] it is proved that t 2 (3, 7) = 17.…”

Section: On the Spectrum Of Sizes 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: Some Complete Caps In Pg(3 Q) With Boundary Sizesmentioning

confidence: 89%

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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“…For small q, better values can often be obtained by determining the exact value or by constructively finding a good upper bound (often by computer search); see [18] and [19, Table 1]. In all but one case, k(2, 4, 1) = 5, the exact value of or the best known upper bound on k(2, q, 1) is attained by a complete cap.…”

Section: On 1-saturating Setsmentioning

confidence: 98%

On Saturating Sets in Small Projective Geometries

Davydov

Östergård²

2000

European Journal of Combinatorics

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A set of points, S ⊆ P G (r, q), is said to be -saturating if, for any point x ∈ P G(r, q), there exist + 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, ). 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) ≤ 3q + 1 for q = 2, q ≥ 4. We further give an upper bound on k ( + 1, p m , ).

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“…Their size k equals 2 N/2 + 2 (N+2)/2 − 3 and almost attains the upper bound in (1.1). For q > 2, q even, apart from small values of both N and q (see [1,6,7,16,19]), the best upper bound on t 2 (N, q), N > 3, is due to Pambianco and Storme [21], who proved that For N = 3, t 2 (3, q) ≤ 3q + 2 was first proved in [23], whereas in [22], it is shown that t 2 (3, q) ≤ 2q + t 2 (2, q) (see also [16, Table 4.8]).…”

Section: Introductionmentioning

confidence: 99%

Small complete caps inPG(N,q),qeven

Giulietti

2006

J of Combinatorial Designs

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Computer search for small complete caps

Cited by 21 publications

References 6 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)

On Saturating Sets in Small Projective Geometries

Small complete caps inPG(N,q),qeven

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