A new (37, 3)-arc in PG(2,23)

Daskalov, Rumen; Manev, Mladen

doi:10.1016/j.endm.2017.02.017

Cited by 4 publications

(21 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We call such an arc a putative arc in PG(2, q). In [6] it was shown that for the gap 100 ≤ m 10 (2, 11) ≤ 101 a putative (101, 10)-arc in PG (2,11) admits-in case of its existence-only the trivial automorphism group of order 1.…”

Section: Excluding Automorphismsmentioning

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%

“…We now apply this corollary to the parameters (q, n, r) = (11, 44, 5), (q, n, r) = (11,90,9), and (q, n, r) = (13, 50, 5). There are | Conj(11)| = 132 conjugacy classes of cyclic subgroups of PGL (3,11) and | Conj(13)| = 184 classes in PGL (3,13). We compute the representatives using GAP [19].…”

mentioning

confidence: 99%

“…By solving the integer linear programming m C r (2, q) according to Theorem 1 for all C ∈ Conj(q) \ {1} using Gurobi [20] we obtain with a runtime less than 3 hours on a 1.2 GHz Intel Core m3 processor the following result: Theorem 2. In case of their existence the automorphism groups of (44, 5)arcs, (90, 9)-arcs in PG (2,11), and (50, 5)-arcs in PG(2, 13) would be trivial of order 1.…”

mentioning

confidence: 99%

See 3 more Smart Citations

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

Braun

2021

Preprint

View full text Add to dashboard Cite

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.

show abstract

Section: Excluding Automorphismsmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

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

Braun

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…In addition, Tables and show lower and upper bound on

m_{r} (2, q)

. For

q goodbreakinfix\leq 19

we refer to the online table “Three‐dimensional linear codes” hosted by Ball (see ) whereas Table for

23 goodbreakinfix\leq q goodbreakinfix\leq 31

is compiled from several sources:

q goodbreakinfix= 23

r goodbreakinfix= 3

q goodbreakinfix= 25

r goodbreakinfix= 24

(, Table 5.1).

q goodbreakinfix= 27

r goodbreakinfix= 26

(, Theorem 3.5).

q goodbreakinfix= 29

q goodbreakinfix= 31

r goodbreakinfix= 21 goodbreakinfix, 22 goodbreakinfix, 23 goodbreakinfix, 24

r goodbreakinfix= 28 goodbreakinfix, 29

r goodbreakinfix= 30

.…”

Section: Introductionmentioning

confidence: 99%

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

Braun¹

2019

J of Combinatorial Designs

View full text Add to dashboard Cite

An (n,r)‐arc in PG(2,q) is a set of n points such that each line contains at most r of the selected points. It is well known that (n,r)‐arcs in PG(2,q) correspond to projective linear codes. Let mr(2,q) denote the maximal number n of points for which an (n,r)‐arc in PG(2,q) exists. In this paper we obtain improved lower bounds on mr(2,q) by explicitly constructing (n,r)‐arcs. Some of the constructed (n,r)‐arcs correspond to linear codes meeting the Griesmer bound. All results are obtained by integer linear programming.

show abstract

Large (k,n)-arcs in the Projective Plane of Order 37

Hanoon

2023

basjs

View full text Add to dashboard Cite

In this study, we use the concept of the cyclic projectivity to find some complete cyclic -arcs that correspond to some divisors of . The cyclic complete arcs , , , and are constructed. Moreover, we find types of large complete -arcs that containing the cyclic arc (469,16), types of large complete -arcs that containing the cyclic arc , types of large complete -arcs that containing the cyclic arc , and types of large complete -arcs that containing the cyclic arc .

show abstract

A new (37, 3)-arc in PG(2,23)

Cited by 4 publications

References 6 publications

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

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

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

Large (k,n)-arcs in the Projective Plane of Order 37

Contact Info

Product

Resources

About