On the spectrum of the valuesk for which a completek- cap in PG(n, q) exists

Faina, Giorgio; Pambianco, Fernanda

doi:10.1007/bf01237602

Cited by 35 publications

(52 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For q ≤ 3701, Vol. 94 (2009) Sizes of complete caps 37 Table 2 The sizes of the known complete k-arcs in PG(2, q) q t 2 (2, q) Sizes k of the known m 2 (2, q) m 2 (2, q) References complete arcs with t 2 (2, q) ≤ k ≤ m 2 (2, q) 43 12 Table 2]. Using the greedy algorithms we obtained complete k-arcs in PG(2, q) with k = 35 for q = 128, k = 80, 82, 83 for q = 163, and k = 83, 84 for q = 167.…”

Section: Construction Cmentioning

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%

See 1 more Smart Citation

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

et al. 2009

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Construction Cmentioning

confidence: 99%

Section: Theoremmentioning

confidence: 99%

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

et al. 2009

Self Cite

View full text Add to dashboard Cite

show abstract

“…For 11 q 32, the spectrum of the sizes of complete arcs in PGð2; qÞ was searched by Chao and Kaneta [3,4,5] and Faina, Marcugini, Milani, and Pambianco [7,8,9,17,18,19,20]. For the range q 19, the number of complete arcs of different lengths in PGð2; qÞ has been tabulated in [12].…”

Section: Types Of Superregular Matricesmentioning

confidence: 99%

“…For the range q 19, the number of complete arcs of different lengths in PGð2; qÞ has been tabulated in [12]. The spectrum of the sizes of complete arcs is known ( [7], [19], [20]) also for q ¼ 23, 25, and 27. The length m 0 ð2; qÞ of the second largest complete arc in PGð2; qÞ is known for q 29 and is monotonously non-decreasing in this range.…”

Section: Types Of Superregular Matricesmentioning

confidence: 99%

“…The question about the number and types of complete MDS codes is also examined and completely (partially) answered for the same pairs of q and k. From this, the known results on the number and structure of complete MDS codes that can be found in [1], [2], [3], [4], [5], [6], [7], [8], [9], [11], [12], [13], [17], [18], [19], [20], and [21] are extended to greater dimensions and to some particular MDS codes of given sizes. Besides, a considerable number of already known results are confirmed.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Types of superregular matrices and the number ofn‐arcs and completen‐arcs in PG (r, q)

Kéri

2005

J of Combinatorial Designs

View full text Add to dashboard Cite

Based on the classification of superregular matrices, the numbers of non-equivalent n-arcs and complete n-arcs in PG(r, q) are determined (i) for 4 q 19, 2 r q À 2 and arbitrary n, (ii) for 23 q 32, r ¼ 2 and n ! q À 8. The equivalence classes over both PGL (k, q) and PÀ ÀL(k, q) are considered throughout the examinations and computations. For the classification, an n-arc is represented by the systematic generator matrix of the corresponding MDS code, without the identity matrix part of it. A rectangular matrix like this is superregular, i.e., it has only non-singular square submatrices. Four types of superregular matrices are studied and the non-equivalent superregular matrices of different types are stored in databases. Some particular results on t(r, q) and m 0 (r, q)-the smallest and the second largest size for complete arcs in PG(r, q)-are also reported, stating that m 0 (2, 31) ¼ 22, m 0 (2, 32) ¼ 24, t(3, 23) ¼ 10, and m 0 (3, 23) ¼ 16. #

show abstract

A full classification of the completek‐arcs of PG(2,23) and PG(2,25)

Coolsaet

Sticker

2009

J of Combinatorial Designs

View full text Add to dashboard Cite

A full classification (up to equivalence) of all complete k-arcs in the Desarguesian projective planes of order 23 and 25 was obtained by computer. The algorithm used is an application of isomorph-free backtracking using canonical augmentation, as introduced by McKay, which we have adapted to the case of subset generation in Desarguesian projective planes. We have applied two variants of the same algorithm, and both techniques yield exactly the same results. Earlier (partial) results by other authors on k-arcs in PG(2, q) with q ≤ 25, are reproduced by our programs. We describe those parts of the algorithms which are relevant to the particular problem of generating k-arcs and which have made this project feasible. We also list the number of complete arcs in PG(2, 23) and PG(2, 25) according to size of the arc and type of the automorphism group. Explicit descriptions are given for the arcs with the larger automorphism groups. q

show abstract

On the spectrum of the valuesk for which a completek- cap in PG(n, q) exists

Cited by 35 publications

References 21 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)

Types of superregular matrices and the number ofn‐arcs and completen‐arcs in PG (r, q)

A full classification of the completek‐arcs of PG(2,23) and PG(2,25)

Contact Info

Product

Resources

About