Complete arcs arising from conics

Korchmáros, Gábor; Sonnino, Angelo

doi:10.1016/s0012-365x(02)00613-1

Cited by 12 publications

(13 citation statements)

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“…Note that for q = 13 the similar contradiction was found in [4]. Note also that for q = 2 p − 1 with odd prime p, plane arcs sharing (q + 3)/2 points with a conic are studied in [19].…”

supporting

confidence: 61%

Complete (q 2 + q + 8)/2-caps in the spaces PG(3, q), q ≡ 2 (mod 3) an odd prime, and a complete 20-cap in PG(3, 5)

Davydov

Marcugini

Pambianco

2008

Des. Codes Cryptogr.

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Complete (q 2 + q + 8)/2-caps in the spaces P G(3, q), q ≡ 2 (mod 3) an odd prime, and a complete 20-cap in P G(3, 5) Abstract An infinite family of complete (q 2 + q + 8)/2-caps is constructed in PG (3, q) where q is an odd prime ≡ 2 ( mod 3), q ≥ 11. This yields a new lower bound on the second largest size of complete caps. A variant of our construction also produces one of the two previously known complete 20-caps in PG(3, 5). The associated code weight distribution and other combinatorial properties of the new (q 2 + q + 8)/2-caps and the 20-cap in PG(3, 5) are investigated. The updated table of the known sizes of the complete caps in PG(3, q) is given. As a byproduct, we have found that the unique complete 14-arc in PG(2, 17) contains 10 points on a conic. Actually, this shows that an earlier general result dating back to the Seventies fails for q = 17.Keywords Complete caps · Projective spaces of the dimension three · Projective planes Mathematics Subject Classifications (2000) 51E21 · 51E22 · 94B05 IntroductionLet PG(n, q) be the projective space of dimension n over the Galois field F q and let F * q = F q \ {0}. A k-cap in PG(n, q) is a set of k points, no three of which are collinear. A k-cap in

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“…Note that for q = 13 the similar contradiction was found in [4]. Note also that for q = 2 p − 1 with odd prime p, plane arcs sharing (q + 3)/2 points with a conic are studied in [19].…”

supporting

confidence: 61%

Complete (q 2 + q + 8)/2-caps in the spaces PG(3, q), q ≡ 2 (mod 3) an odd prime, and a complete 20-cap in PG(3, 5)

Davydov

Marcugini

Pambianco

2008

Des. Codes Cryptogr.

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“…Replacing P (or Q) in the above argument shows that K 0 is also preserved by two more linear collineations of order two. Now, an argument depending on the classification of subgroups of PGL(2, q) as developed in [15,Section 4] can be applied which actually rules out the existence of such a point R under our hypothesis q 2 ≡ 1 (mod 16).…”

Section: {O (O) (Q+1)/2 (O) ( (Q+1)/2 )(O)}mentioning

confidence: 99%

“…In [15], see also [14], the case where 1 2 (q +1) is a prime was settled showing that n ≤ 4. In Section 5, a stricter result is proven for the case where 1 4 (q +1) is prime; see Theorem 5.1.…”

mentioning

confidence: 99%

On arcs sharing the maximum number of points with ovals in cyclic affine planes of odd order

Korchmáros¹,

Sonnino²

2009

J of Combinatorial Designs

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The sporadic complete $12$-arc in $\mathrm{PG}(2,13)$ contains eight points from a conic. In $\mathrm{PG}(2,q)$ with $q>13$ odd, all known complete $k$-arcs sharing exactly $\ha(q+3)$ points with a conic $\mathcal{C}$ have size at most $\frac{1}{2}(q+3)+2$, with only two exceptions, both due to Pellegrino, which are complete $(\frac{1}{2}(q+3)+3)$ arcs, one in $\mathrm{PG}(2,19)$ and another in $\mathrm{PG}(2,43)$. Here, three further exceptions are exhibited, namely a complete $(\frac{1}{2}(q+3)+4)$-arc in $\mathrm{PG}(2,17)$, and two complete $(\frac{1}{2}(q+3)+3)$-arcs, one in $\mathrm{PG}(2,27)$ and another in $\mathrm{PG}(2,59)$. The main result is Theorem 6.1 which shows the existence of a $(\frac{1}{2}(q^r+3)+3)$--arc in $\mathrm{PG}(2,q^r)$ with $r$ odd and $q\equiv 3 \pmod 4$ sharing $\frac{1}{2}(q^r+3)$ points with a conic, whenever $\mathrm{PG}(2,q)$ has a $(\frac{1}{2}(q+3)+3)$-arc sharing $\frac{1}{2}(q+3)$ points with a conic. A survey of results for smaller $q$ obtained with the use of the MAGMA package is also presented

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“…Arcs different from conics, but sharing as many points as possible with conics, were also studied in [5,14,22,23]. Currently, k-arcs in the Desarguesian projective plane PG(2, q) have been investigated, not only from a theoretical point of view, but also for their natural applications to coding theory and cryptography; see for instance [11,12,17,18,21,27,28,32].…”

Section: Introductionmentioning

confidence: 99%