The Complete Arcs of PG(2,31)

Coolsaet, Kris

doi:10.1002/jcd.21410

Cited by 8 publications

(8 citation statements)

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“…Theorem 1.1, Theorem 1.2 and Theorem 1.4 may be of some use in classifying or at least constructing large arcs computationally. See [6], [7] and [10] for recent computational results regarding arcs. To classify arcs of size q + k − r one would need to classify arcs of size k + r. If one could classify arcs of size k + r then one could quickly check for each arc to see if M 1 has a vector of weight one in the column space, for each projectively distinct arc G. In the positive case, this would then rule out the possibility that G can be extended to an arc of size q + k − r. In the negative case, one can then extend the arc to an arc H of size k + r + 1 and check to see if M 2 (calculated using H) has a vector of weight one in the column space.…”

Section: Introductionmentioning

confidence: 99%

Extending small arcs to large arcs

Ball

2017

European Journal of Mathematics

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An arc is a set of vectors of the k-dimensional vector space over the finite field with q elements Fq, in which every subset of size k is a basis of the space, i.e. every k-subset is a set of linearly independent vectors. Given an arc G in a space of odd characteristic, we prove that there is an upper bound on the largest arc containing G. The bound is not an explicit bound but is obtained by computing properties of a matrix constructed from G. In some cases we can also determine the largest arc containing G, or at least determine the hyperplanes which contain exactly k − 2 vectors of the large arc. The theorems contained in this article may provide new tools in the computational classification and construction of large arcs. The article also simplifies some of the proofs of the results found in [1], [2, Chapter 7], [3] and [5] and unifies the approach taken in those articles with that in [4] and [13]. u∈G\E det(u, C), if and only if A ⊂ C, and zero otherwise.We shall prove the following theorem.2000 Mathematics Subject Classification 51E21, 15A03, 94B05, 05B35.

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Section: Introductionmentioning

confidence: 99%

Extending small arcs to large arcs

Ball

2017

European Journal of Mathematics

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“…In (2, ) for a certain value of . For =2 see (22)(23)(24)(25), for =3 see (26)(27)(28), and for n ≥ 4 see (29-31).…”

Section: Literature Reviewmentioning

confidence: 99%

A complete (48, 4)-arc in the Projective Plane Over the Field of Order Seventeen

Hamed

Hirschfeld

2021

Baghdad Sci.J

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The article describes a certain computation method of ( , )-arcs to construct the number of distinct ( , 4)-arcs in PG(2,17) for = 7, … ,48. In this method, a new approach employed to compute the number of ( , )-arcs and the number of distinct ( , )-arcs respectively. This approach is based on choosing the number of inequivalent classes { 4 , 3 , 2 , 1 , 0 } of -secant distributions that is the number of 4-secant, 3secant, 2-secant, 1-secant and 0-secant in each process. The maximum size of ( , 4)-arc that has been constructed by this method is = 48. The new method is a new tool to deal with the programming difficulties that sometimes may lead to programming problems represented by the increasing number of arcs. It is essential to reduce the established number of ( , )-arcs in each construction especially for large value of and then reduce the running time of the calculation. Therefore, it allows to decrease the memory storage for the calculation processes. This method's effectiveness evaluation is confirmed by the results of the calculation where a largest size of complete ( , 4)-arc is constructed. This research's calculation results develop the strategy of the computational approaches to investigate big sizes of ( , )-arcs in (2, ) where it put more attention to the study of the number of the inequivalent classes of -secants of ( , )-arcs in (2, ) which is an interesting aspect. Consequently, it can be used to establish a large value of .

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“…For q odd the situation is quite different. The combination of computer aided calculations (see for example the articles by Coolsaet and Sticker [9,10] and Coolsaet [8]) with theoretical upper bounds (see Section 2) were not enough to complete the classification of complete arcs of size q − 1.…”

Section: Bounds On Arcs Contained In Low Degree Curvesmentioning

confidence: 99%

“…According to [14,Theorem 10.33], there are 15 values of q, all odd and satisfying q 31 (q = 31 has since been ruled out by Coolsaet [8]), for which it is not yet determined whether an arc of size q − 1 is complete or not. We can now complete the classification of planar arcs of size q − 1.…”

Section: Bounds On Arcs Contained In Low Degree Curvesmentioning

confidence: 99%

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Planar arcs

Ball

Lavrauw

2018

Journal of Combinatorial Theory, Series A

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Let p denote the characteristic of F q , the finite field with q elements. We prove that if q is odd then an arc of size q + 2 − t in the projective plane over F q , which is not contained in a conic, is contained in the intersection of two curves, which do not share a common component, and have degree at most t + p ⌊log p t⌋ , provided a certain technical condition on t is satisfied.This implies that if q is odd then an arc of size at least q − √ q + √ q/p + 3 is contained in a conic if q is square and an arc of size at least q − √ q + 7 2 is contained in a conic if q is prime. This is of particular interest in the case that q is an odd square, since then there are examples of arcs, not contained in a conic, of size q − √ q + 1, and it has long been conjectured that if q = 9 is an odd square then any larger arc is contained in a conic.These bounds improve on previously known bounds when q is an odd square and for primes less than 1783. The previously known bounds, obtained by Segre [23], Hirschfeld and Korchmáros [15] [16], and Voloch [29] [30], rely on results on the number of points on algebraic curves over finite fields, in particular the Hasse-Weil theorem and the Stöhr-Voloch theorem, and are based on Segre's idea to associate an algebraic curve in the dual plane containing the tangents to an arc. In this paper we do not rely on such theorems, but use a new approach starting from a scaled coordinate-free version of Segre's lemma of tangents.Arcs in the projective plane over F q of size q and q + 1, q odd, were classified by Segre [22] in 1955. In this article, we complete the classification of arcs of size q − 1 and q − 2.

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The Complete Arcs of PG(2,31)

Cited by 8 publications

References 11 publications

Extending small arcs to large arcs

Extending small arcs to large arcs

A complete (48, 4)-arc in the Projective Plane Over the Field of Order Seventeen

Planar arcs

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