Projective k-arcs and 2-level secret-sharing schemes

Korchmáros, Gábor; Lanzone, Valentino; Sonnino, Angelo

doi:10.1007/s10623-011-9562-5

Cited by 15 publications

(6 citation statements)

References 12 publications

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“…Cherowitzo and Holder [4] classified all hyperfocused arcs contained in a regular hyperoval. In [14] showed that this classification extends to every translation hyperoval Ω provided that ℓ is the special chord of Ω. Faina and Pasticci [6] proved that in small planes and for special values of k, every hyperfocused k-arc is contained in a regular hyperoval. Nevertheless, Giulietti and Montanucci [9] constructed a family of hyperfocused arcs not contained in any hyperoval.…”

Section: Hyperfocused Arcs In P G(2 Q)mentioning

confidence: 96%

Problems and results in

Korchmáros

2013

Electronic Notes in Discrete Mathematics

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Section: Hyperfocused Arcs In P G(2 Q)mentioning

confidence: 96%

Problems and results in

Korchmáros

2013

Electronic Notes in Discrete Mathematics

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“…Of course the two words in italics need to be given a precise meaning. A very special case of this occurred in [KLS12,Lemma 5.3], where the conclusion was proved under the assumption that A is inverse closed up to at most two nonzero elements.…”

Section: Introductionmentioning

confidence: 98%

Inversion and subspaces of a finite field

Mattarei

2014

Isr. J. Math.

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Consider two F q -subspaces A and B of a finite field, of the same size, and let A −1 denote the set of inverses of the nonzero elements of A. The author proved that A −1 can only be contained in A if either A is a subfield, or A is the set of trace zero elements in a quadratic extension of a field. Csajbók refined this to the following quantitative statement: if A −1 ⊈ B, then the bound |A −1∩B| ≤ 2|B|/q − 2 holds. He also gave examples showing that his bound is sharp for |B| ≤ q 3. Our main result is a proof of the stronger bound |A −1 ∩ B| ≤ |B|/q · (1 + O d (q −1/2)), for |B| = q d with d > 3. We also classify all examples with |B| ≤ q 3 which attain equality or near-equality in Csajbók’s bound

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“…codes and k-arcs are equivalent objects; see [16]. For more on the connection between arcs and secret sharing schemes, see, for instance, [9], [10], and [14]. For these reasons, in recent years there has been great interest in constructing new arcs in projective spaces; see [2], [7], [8], [11], [12], [13], [15].…”

Section: Introductionmentioning

confidence: 99%

Transitive PSL(2,11)-invariant k-arcs in PG(4,q)

Olson

Swartz

2018

Des. Codes Cryptogr.

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A k -arc in the projective space PG(n, q) is a set of k projective points such that no subcollection of n + 1 points is contained in a hyperplane. In this paper, we construct new 60-arcs and 110-arcs in PG(4, q) that do not arise from rational or elliptic curves. We introduce computational methods that, when given a set P of projective points in the projective space of dimension n over an algebraic number field Q(ξ), determines a complete list of primes p for which the reduction modulo p of P to the projective space PG(n, p h ) may fail to be a k-arc. Using these methods, we prove that there are infinitely many primes p such that PG(4, p) contains a PSL(2, 11)-invariant 110-arc, where PSL(2, 11) is given in one of its natural irreducible representations as a subgroup of PGL (5, p). Similarly, we show that there exist PSL(2, 11)-invariant 110-arcs in PG(4, p 2 ) and PSL(2, 11)-invariant 60-arcs in PG(4, p) for infinitely many primes p.

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Projective k-arcs and 2-level secret-sharing schemes

Cited by 15 publications

References 12 publications

Problems and results in

Problems and results in

Inversion and subspaces of a finite field

Transitive PSL(2,11)-invariant k-arcs in PG(4,q)

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