New inductive constructions of complete caps inPG(N, q),qeven

Davydov, Alexander A.; Giulietti, Massimo; Marcugini, Stefano; Pambianco, Fernanda

doi:10.1002/jcd.20230

Cited by 28 publications

(48 citation statements)

References 20 publications

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“…The updated table of t 2 (2, q) is given as Table 1. For q = 2 7 a complete 34-arc is obtained from the affinely complete 32-arc of [48, Appendix, Lemma 4.3] by adding two points; see also [21,Section 2]. From Table 1, we obtain Theorem 1, improving the corresponding theorem in [17, p. 55].…”

Section: Complete Arcs In Planes Pg(2 Q)supporting

confidence: 50%

“…For even q = 2 h , 10 ≤ h ≤ 15, the smallest known sizes of complete k-arcs in PG(2, q) are obtained in [21]; they are as follows: k = 124, 201, 307, 461, 665, 993, for h = 10, 11, 12, 13, 14, 15, respectively. Also, 6( √ q − 1)-arcs in PG(2, q), q = 4 2h+1 , are constructed in [22]; for h ≤ 4 it is proved that they are complete.…”

Section: Complete Arcs In Planes Pg(2 Q)mentioning

confidence: 99%

“…The unique 18-arc K 27 (3) may be represented as follows: (1,12,23), (1,10,19); (1, 0, 0), (0, 0, 1), (1, 2, 3), (1,22,17), (1,13,25), (1,11,21); (1,8,15), (1,20,13), (1,19,11), (1,16,5), (1,4,7), (1,5,9) It should be noted that, independently of this work, in the recent paper [61], the complete arcs K 17 (4), K 19 (3), K 27 (3), K 43 (3), and K 59 (3) are obtained with the help of an interesting theoretical approach supported by computer search. Moreover, in [61, Theorem 6.1] infinite families of ( 1 2 (q + 3) + δ)-arcs are constructed for q ≡ 3 (mod 4).…”

Section: Theoremmentioning

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: Complete Arcs In Planes Pg(2 Q)supporting

confidence: 50%

Section: Complete Arcs In Planes Pg(2 Q)mentioning

confidence: 99%

Section: Theoremmentioning

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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“…From (8), it follows that e γ 1 = 1; then v γ 1 (ū) = −1 is obtained from (7). As far as γ 2 is concerned, note that…”

Section: Lemmamentioning

confidence: 96%

“…Otherwise, all known infinite families of complete caps have size far from (1); see the survey papers [17,18] and the more recent works [1,4,5,7,8,[12][13][14]. For q odd and N = 2, the smallest explicit constructions go back to the late 80's, when Szőnyi described complete plane arcs of size approximately (q − 1)/m for any divisor m of q − 1 smaller than 1 C q 1/4 , with C a constant independent of q and greater than 1 [27,28] 1 .…”

Section: Introductionmentioning

confidence: 99%