Small complete caps from singular cubics, II

Anbar, Nurdagül; Bartoli, Daniele; Platoni, Irene; Giulietti, Massimo

doi:10.1007/s10801-014-0532-7

Cited by 14 publications

(6 citation statements)

References 22 publications

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“…Assume that

m = m_{1} m_{2}

with

(m_{1}, m_{2}) = 1

. Then for

N \equiv 0 mod 4

N \geq 4

, there exists a complete cap in

A G (N, q)

of size at most

\frac{m_{1} + m_{2}}{m} false(q - 1 false) q^{\frac{N - 2}{2}} .

[, Theorem 6] Let

q = p^{h}

p > 3

prime, and let m be a proper divisor of

q + 1

such that

(m, 6) = 1

and

m \leq \sqrt[4]{q} / 4

. Assume that

m = m_{1} m_{2}

with

(m_{1}, m_{2}) = 1

.…”

Section: Small Complete Caps In Ag(nq) N and Q Oddmentioning

confidence: 99%

“…Complete caps of size roughly

2 p^{β} q^{(4 N - 1) / 8}

, with

β \in [1 / 8, 1]

and p the characteristic of the ground field

F_{q}

, have been described in [, Theorem 6.2]. For a divisor m of

q + 1

q - 1

with

(m, 6) = 1

and

m \leq \sqrt[4]{q} / 4

, such that m admits a nontrivial factorization

m = m_{1} m_{2}

with

(m_{1}, m_{2}) = 1

, complete caps with roughly

\frac{m_{1} + m_{2}}{m} q^{N / 2}

points in affine spaces

A G (N, q)

with dimension

N \equiv 0 (mod 4)

are provided in . All these constructions rely on the notion of bicovering and almost bicovering arcs in affine planes

A G (2, q)

; see also where some computer‐assisted constructions of both bicovering and almost bicovering arcs in

A G (2, q)

for small q 's are provided.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Complete Caps in AG(N,q) with BothNandqOdd

et al. 2017

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“…Assume that

m = m_{1} m_{2}

with

(m_{1}, m_{2}) = 1

. Then for

N \equiv 0 mod 4

N \geq 4

, there exists a complete cap in

A G (N, q)

of size at most

\frac{m_{1} + m_{2}}{m} false(q - 1 false) q^{\frac{N - 2}{2}} .

[, Theorem 6] Let

q = p^{h}

p > 3

prime, and let m be a proper divisor of

q + 1

such that

(m, 6) = 1

and

m \leq \sqrt[4]{q} / 4

. Assume that

m = m_{1} m_{2}

with

(m_{1}, m_{2}) = 1

.…”

Section: Small Complete Caps In Ag(nq) N and Q Oddmentioning

confidence: 99%

“…Complete caps of size roughly

2 p^{β} q^{(4 N - 1) / 8}

, with

β \in [1 / 8, 1]

and p the characteristic of the ground field

F_{q}

, have been described in [, Theorem 6.2]. For a divisor m of

q + 1

q - 1

with

(m, 6) = 1

and

m \leq \sqrt[4]{q} / 4

, such that m admits a nontrivial factorization

m = m_{1} m_{2}

with

(m_{1}, m_{2}) = 1

, complete caps with roughly

\frac{m_{1} + m_{2}}{m} q^{N / 2}

points in affine spaces

A G (N, q)

with dimension

N \equiv 0 (mod 4)

are provided in . All these constructions rely on the notion of bicovering and almost bicovering arcs in affine planes

A G (2, q)

; see also where some computer‐assisted constructions of both bicovering and almost bicovering arcs in

A G (2, q)

for small q 's are provided.…”

Section: Introductionmentioning

confidence: 99%

Complete Caps in AG(N,q) with BothNandqOdd

et al. 2017

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“…Concerning the smallest interesting case m = 2, the theory is well developed and quite rich of constructions; see e.g. [1,2,3,19,23,35,38] and the references therein, as well as [20,. The notion of arcs (with m = 2) has been introduced by Segre [31,32] in the late 1950's.…”

Section: Introductionmentioning

confidence: 99%

Algebraic constructions of complete $m$-arcs

Bartoli¹,

Micheli²

2020

Preprint

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Let m be a positive integer, q be a prime power, and PG(2, q) be the projective plane over the finite field Fq. Finding complete m-arcs in PG(2, q) of size less than q is a classical problem in finite geometry. In this paper we give a complete answer to this problem when q is relatively large compared with m, explicitly constructing the smallest m-arcs in the literature so far for any m ≥ 8. For any fixed m, our arcs Aq,m satisfy |Aq,m| − q → −∞ as q grows. To produce such m-arcs, we develop a Galois theoretical machinery that allows the transfer of geometric information of points external to the arc, to arithmetic one, which in turn allows to prove the m-completeness of the arc.

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“…By similar methods, in [10] the authors provide new complete caps in AG(N, q) with roughly q (4N −1)/8 points, studying both plane cubics with a node and plane cubics with an isolated double point.…”

Section: Introductionmentioning

confidence: 99%

A construction of small complete caps in projective spaces

et al. 2016

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In this work complete caps in P G(N, q) of size O(q N −1 2 log 300 q) are obtained by probabilistic methods. This gives an upper bound asymptotically very close to the trivial lower bound √ 2q N −1 2 and it improves the best known bound in the literature for small complete caps in projective spaces of any dimension. The result obtained in the paper also gives a new upper bound for l(m, 2, q) 4 , that is the minimal length n for which there exists an [n, n − m, 4] q 2 covering code with given m and q.

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Small complete caps from singular cubics, II

Abstract: Small complete arcs and caps in Galois spaces over finite fields F q with characteristic greater than three are constructed from singular cubic curves.

Cited by 14 publications

References 22 publications

Complete Caps in AG(N,q) with BothNandqOdd

Complete Caps in AG(N,q) with BothNandqOdd

Algebraic constructions of complete $m$-arcs

A construction of small complete caps in projective spaces

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