The complete k-arcs of PG(2, 27) and PG(2, 29)

Coolsaet, Kris; Sticker, Heide

doi:10.1002/jcd.20261

Cited by 10 publications

(25 citation statements)

References 15 publications

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“…It was already proved in that

S (a)

(

= S (- a)

) is a 12‐arc of PG(2, q ) if and only if

a \notin \{0, \pm 1, \pm 2, \pm \sqrt{- 1}, \pm \sqrt{- 3}, \frac{1}{2} (\pm 1 \pm \sqrt{- 7})\} .

…”

Section: Complete Arcs Of Size 20 With Gs≃s4mentioning

confidence: 95%

“…We verified by computer that applying Theorem 2.1 to the pairs (A, B) listed above, and choosing the largest of S 1 (A, B), S 2 (A, B), or S 12 (A, B) in each case, yields an arc that is complete, except for the three small cases (q, A, B) = (7, 1, −1), (7,1,3), and (13,1,5).…”

Section: Arcs On Two Conicsmentioning

confidence: 99%

“…In earlier publications [, ] we have reported on exhaustive computer searches for arcs in planes of order 23, 25, 27, and 29. In this paper, we present the results for

q = 31

.…”

Section: Introductionmentioning

confidence: 99%

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The Complete Arcs of PG(2,31)

Coolsaet

2014

J. Combin. Designs

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We obtained a full computer classification of all complete arcs in the Desarguesian projective plane of order 31 using essentially the same methods as for earlier results for planes of smaller order, i.e., isomorph‐free backtracking using canonical augmentation. We tabulate the resulting numbers of complete arcs according to size and automorphism group. We give explicit descriptions for all complete arcs with an automorphism group of size at least 20. In some of these cases the constructions can be generalized to other values of q. In particular, we find arcs of size 20 for any field of order q=1(mod6), q≥31 and a complete 44‐arc in PG(2,67) with an automorphism group of order 88. We also correct a result by Kéri : there are 12 complete 22‐arcs in PG(2,31) up to projective equivalence, and not 11.

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“…It was already proved in that

S (a)

(

= S (- a)

) is a 12‐arc of PG(2, q ) if and only if

a \notin \{0, \pm 1, \pm 2, \pm \sqrt{- 1}, \pm \sqrt{- 3}, \frac{1}{2} (\pm 1 \pm \sqrt{- 7})\} .

…”

Section: Complete Arcs Of Size 20 With Gs≃s4mentioning

confidence: 95%

Section: Arcs On Two Conicsmentioning

confidence: 99%

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The Complete Arcs of PG(2,31)

Coolsaet

2014

J. Combin. Designs

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“…If h i = 0, for i ≤ 9, then all expressions in (1) are nonzero. In (6,7,8), there are nine linear expressions, 12 expressions of degree 2, and six expressions of degree 3. Then, there are at most 51 values of a ∈ GF (q) such that h i (a) = 0, for some i.…”

Section: Case P =mentioning

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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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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The complete (k, 3)‐arcs of PG(2,q),q≤13

Coolsaet

Sticker

2011

J of Combinatorial Designs

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We have classified by computer the projectively distinct complete (k, 3)-arcs in PG(2, q), q ≤ 13. The algorithm used is an application of isomorphfree backtracking using canonical augmentation, an adaptation of our earlier algorithms for the generation of (k, 2)-arcs. We describe those parts of the algorithms which are specific to the particular problem of (k, 3)-arcs. For each of these arcs we have also determined the automorphism group. The results are summarised in tables where the arcs are listed according to size and automorphism group. For the arcs with the larger automorphism groups, explicit descriptions are given. Part of the computer results can be generalized to other values of q: we describe constructions of arcs having S 4 as a group of automorphisms, arcs containing the union of three 'half conics' and arcs constructed from parts of cubic curves.

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The complete k-arcs of PG(2, 27) and PG(2, 29)

Cited by 10 publications

References 15 publications

The Complete Arcs of PG(2,31)

The Complete Arcs of PG(2,31)

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

The complete (k, 3)‐arcs of PG(2,q),q≤13

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