New lower bounds on the size of (n,r)‐arcs in PG(2,q)

Braun, Michael

doi:10.1002/jcd.21672

Cited by 8 publications

(22 citation statements)

References 12 publications

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“…In this article, we extend the results from [5] of lower bounds on m r (2, q) and give some improvements listed in Table 1.…”

mentioning

confidence: 84%

“…We use the construction of (n, r)-arcs in PG(2, q) with prescribed groups of automorphisms using integer linear programming described in [4,5]:…”

Section: Construction By Integer Linear Programmingmentioning

confidence: 99%

“…The values m r (2, q) with q ≤ 9 are exactly determined (see [3]). For m r (2, q) with 11 ≤ q ≤ 19 we refer to [2] whereas a table for 23 ≤ q ≤ 31 can compiled from several sources [5,7,8,9,10,11,12,13,14,15,16,17,18]. An recent overview with tables on all values q ≤ 31 can be found in [5].…”

mentioning

confidence: 99%

“…, 0, 4), (1,1,17), (1,15,10), (1,15,19), (1,0,6), (1,3,13), (1,2,3), (1,5,21), (1, 0, 8), (1,21,11), (1,6,24), (1,4,22), (1, 0, 10), (1,4,21), (1,13,17), (1, 7, 1), (1,0,11), (1,15,22), (1, 3, 2), (1,21,18), (1,0,14), (1,17,18), (1,…”

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confidence: 99%

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Update: Some new results on lower bounds on $(n,r)$-arcs in $PG(2,q)$ for $q\le 31$

Braun

2021

Preprint

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An (n, r)-arc in PG(2, q) is a set B of points in PG(2, q) such that each line in PG(2, q) contains at most r elements of B and such that there is at least one line containing exactly r elements of B. The value m r (2, q) denotes the maximal number n of points in the projective geometry PG(2, q) for which an (n, r)-arc exists. By explicitly constructing (n, r)-arcs using prescribed automorphisms and integer linear programming we obtain some improved lower bounds for m r (2, q): m 10 (2, 16) ≥ 144, m 3 (2, 25) ≥ 39, m 18 (2, 25) ≥ 418, m 9 (2, 27) ≥ 201, m 14 (2, 29) ≥ 364, m 25 (2, 29) ≥ 697, m 25 (2, 31) ≥ 734. Furthermore, we show by systematically excluding possible automorphisms that putative (44, 5)-arcs, (90, 9)-arcs in PG(2, 11), and (39, 4)-arcs in PG(2, 13)-in case of their existence-are rigid, i.e. they all would only admit the trivial automorphism group of order 1. In addition, putative (50, 5)-arcs, (65, 6)-arcs, (119, 10)-arcs, (133, 11)-arcs, and (146, 12)-arcs in PG(2, 13) would be rigid or would admit a unique automorphism group (up to conjugation) of order 2.

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“…In this article, we extend the results from [5] of lower bounds on m r (2, q) and give some improvements listed in Table 1.…”

mentioning

confidence: 84%

“…We use the construction of (n, r)-arcs in PG(2, q) with prescribed groups of automorphisms using integer linear programming described in [4,5]:…”

Section: Construction By Integer Linear Programmingmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Update: Some new results on lower bounds on $(n,r)$-arcs in $PG(2,q)$ for $q\le 31$

Braun

2021

Preprint

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“…So, the number of points and lines in the plane (2,17) is 307, with 18 points on each line and 18 lines concurrent with a point as given in Tables 1 and 2. (11)(12)(13). The answers to these questions are given in Tables 3-23.…”

Section: Introductionmentioning

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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