Infinite family of large complete arcs in PG(2, q n ), with q odd and n &gt; 1 odd

Korchmáros, Gábor; Pace, Nicola

doi:10.1007/s10623-009-9343-6

Cited by 10 publications

(37 citation statements)

References 16 publications

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“…q = 97 : There exist complete 28-arcs that are the union of two S 4 -orbits. For example, the points P 4 (1, 96, 96), P 24 (1, 36, 90) can be considered. Then, we have that |S(P 4 )| = 4, |S(P 24 )| = 24, and S(P 4 ) ∪ S(P 24 ) is a complete 28-arc in P G(2, 97).…”

Section: Computational Results For the Cases G ∼ = A 4 And G ∼ = Smentioning

confidence: 99%

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On Small Complete Arcs and Transitive A5-Invariant Arcs in the Projective Plane PG(2,q)

Pace

2013

J. Combin. Designs

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Let q be an odd prime power such that q is a power of 5 or q≡±1 (mod 10). In this case, the projective plane PG(2,q) admits a collineation group G isomorphic to the alternating group A5. Transitive G‐invariant 30‐arcs are shown to exist for every q≥41. The completeness is also investigated, and complete 30‐arcs are found for q=109,121,125. Surprisingly, they are the smallest known complete arcs in the planes PG(2,109),PG(2,121), and PG(2,125). Moreover, computational results are presented for the cases G≅A4 and G≅S4. New upper bounds on the size of the smallest complete arc are obtained for q=67,97,137,139,151.

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Section: Computational Results For the Cases G ∼ = A 4 And G ∼ = Smentioning

confidence: 99%

“…The projectivity group which preserves a

(q + 1)

–arc

scriptK

P G (2, q)

, with q odd, can be shown to be isomorphic to the projective linear group

P G L (2, q)

and acts on

scriptK

P G L (2, q)

in its natural 3–transitive permutation representation. Arcs with many projectivities were investigated in several papers, see .…”

Section: Introductionmentioning

confidence: 99%

On Small Complete Arcs and Transitive A5-Invariant Arcs in the Projective Plane PG(2,q)

Pace

2013

J. Combin. Designs

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“…Hence P 1 = (1, 5, 11). A direct computer-aided computation shows that δ i, j = 0 for the twenty-four triples {P 1 , P i , P j } with [i, j] ranging over the set { [3,11], [3,20], [6,9], [6,19], [8,35], [8,41], [9,19], [11,20], [12,13], [12,21], [13,21], [14,16], [14,18], [15,22], [15,24], [16,18], [22,24], [23,34], …”

Section: P = 29mentioning

confidence: 99%

“…Hence P 1 = (w 12432 , w 10752 , 1). A direct computer-aided computation shows that δ i, j = 0 for each of the 24 triples {P 1 , P i , P j } with [i, j] ranging over the set { [4,20], [4,22], [7,17], [7,29], [9,30], [9,33], [10,13], [10,37], [11,24], [11,32], [13,37], [16,27], [16,40], [17,29], [18,34], [18,42], [20,22] …”

Section: P =mentioning

confidence: 99%

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42-arcs in PG(2, q) left invariant by PSL(2, 7)

Indaco

Korchmáros

2011

Des. Codes Cryptogr.

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For an odd prime p = 7, let q be a power of p such that q3 ≡ 1 (mod 7). It is known that the desarguesian projective plane PG(2, q) of order q has a unique conjugacy class of projectivity groups isomorphic to PSL(2, 7). For such a projective group , we\ud investigate the geometric properties of the (unique) -orbit of size 42 such that the 1-point stabilizer of in is a cyclic group of order 4. We present a computational approach to prove that is a 42-arc provided that q ≥ 53 and q = 373, 116, 56, 36. We discuss the case q = 53 in more detail showing the completeness of for q = 53

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Transitive A 6-invariant k-arcs in PG(2, q)

Giulietti

Korchmáros

Marcugini

et al. 2012

Des. Codes Cryptogr.

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For q = p r with a prime p ≥ 7 such that q ≡ 1 or 19 (mod 30), the desarguesian projective plane P G(2, q) of order q has a unique conjugacy class of projectivity groups isomorphic to the alternating group A6 of degree 6. For a projectivity group Γ ∼ = A6 of P G(2, q), we investigate the geometric properties of the (unique) Γ-orbit O of size 90 such that the 1-point stabilizer of Γ in O is a cyclic group of order 4. Here O lies either in P G(2, q) or in P G(2, q 2 ) according as 3 is a square or a non-square element in GF (q). We show that if q ≥ 349 and q = 421, then O is a 90-arc, which turns out to be complete for q = 349, 409, 529, 601, 661. Interestingly, O is the smallest known complete arc in P G(2, 601) and in P G(2, 661). Computations are carried out by MAGMA.

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Infinite family of large complete arcs in PG(2, q n ), with q odd and n > 1 odd

Cited by 10 publications

References 16 publications

On Small Complete Arcs and Transitive A5-Invariant Arcs in the Projective Plane PG(2,q)

On Small Complete Arcs and Transitive A5-Invariant Arcs in the Projective Plane PG(2,q)

42-arcs in PG(2, q) left invariant by PSL(2, 7)

Transitive A 6-invariant k-arcs in PG(2, q)

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