Large caps

Bierbrauer, Jürgen

doi:10.1007/s00022-003-1690-2

Cited by 22 publications

(26 citation statements)

References 23 publications

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“…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: Theoremmentioning

confidence: 99%

“…see [52,Theorem 5.5], [57, Table 4.5] for (30) and [12] for (31). The complete k-caps with k = 56 in PG(4, 7), k = 114 in PG (6,4), and k = 534 in PG (8,3) noted by the bold font in Table 6 are obtained by computer in this work with the help of the randomized greedy algorithms [17,23].…”

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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“…Then, for every constant k ∈ N and constant ε ∈ (0, 0.5), there exists an explicit set S ⊆ F n of size |F| (1−ε)n which is k, 2 ε k -subspace evasive. 1 We stress that in the setting considered by Guruswami [3] both k and ε are constants, and the field size is polynomial in n. Thus, using Theorem 1.2, we get explicit codes of length n and rate R which are (1 − R − 2ε, 2 ε 1/ε ) list-decodable. Moreover, they can be encoded and list-decoded in time polynomial in n.…”

Section: Explicit Subspace Evasive Setsmentioning

confidence: 92%

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Ben-Aroya¹,

Shinkar²

2014

Chicago J. of Theoretical Comp. Sci.

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A subspace evasive set over a field F is a subset of F n that has small intersection with any low-dimensional affine subspace of F n . Interest in subspace evasive sets began in the work of Pudlák and Rödl (Quaderni di Matematica 2004) in the context of explicit constructions of Ramsey graphs. More recently, Guruswami (CCC 2011) showed that obtaining such sets over large fields can be used to construct capacity-achieving list-decodable codes with a constant list size.Our results in this note are as follows:• We provide a construction of subspace evasive sets in F n of size |F| (1−ε)n that intersect any k-dimensional affine subspace of F n in at most (2/ε) k points. This slightly improves a recent construction of Dvir and Lovett (STOC 2012) in terms of the intersection size, who constructed similar sets, but with a bound of (k/ε) k on the size of the intersection. Besides having a smaller intersection, our construction is more elementary. The construction is explicit when k and ε are constants. This is sufficient in order to explicitly construct the aforementioned list-decodable codes. On the other hand, the construction of Dvir and Lovett is explicit in a stronger sense, and relies on solving systems of polynomial equations, while our construction relies on the method of conditional expectation.• We use Kövári-Sós-Turán Theorem to show that for a certain range of parameters the subspace evasive sets obtained using the probabilistic method are optimal (up to a multiplicative constant factor).

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“…[2,7,14]). On the other hand, again, the caps resulting from the sequences for r = 6, 7 are complete in AG(r, 3).…”

Section: S| =mentioning

confidence: 97%

Sequences in abelian groups G of odd order without zero-sum subsequences of length exp(G)

Edel

2007

Des. Codes Cryptogr.

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Large caps

Cited by 22 publications

References 23 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)

Untitled

Sequences in abelian groups G of odd order without zero-sum subsequences of length exp(G)

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