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A linear set view on KM-arcs
In this paper, we study KM-arcs of type t , i.e., point sets of size q + t in PG ( 2 , q ) such that every line contains 0, 2 or t of its points. We use field reduction to give a different point of view on the class of translation arcs. Starting from a particular F 2 -linear set, called an i -club ,...
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| Published in: | Journal of algebraic combinatorics 2016-08, Vol.44 (1), p.131-164 |
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| Main Authors: | , |
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| Language: | English |
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| container_start_page | 131 |
| container_title | Journal of algebraic combinatorics |
| container_volume | 44 |
| creator | De Boeck, Maarten Van de Voorde, Geertrui |
| description | In this paper, we study KM-arcs of type
t
, i.e., point sets of size
q
+
t
in
PG
(
2
,
q
)
such that every line contains 0, 2 or
t
of its points. We use field reduction to give a different point of view on the class of
translation
arcs. Starting from a particular
F
2
-linear set, called an
i
-club
, we reconstruct the
projective triads
, the
translation hyperovals
as well as the translation arcs constructed by Korchmáros-Mazzocca, Gács-Weiner and Limbupasiriporn. We show the KM-arcs of type
q
/
4
recently constructed by Vandendriessche are translation arcs and fit in this family. Finally, we construct a family of KM-arcs of type
q
/
4
. We show that this family, apart from new examples that are not translation KM-arcs, contains all translation KM-arcs of type
q
/
4
. |
| doi_str_mv | 10.1007/s10801-015-0661-7 |
| format | article |
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t
, i.e., point sets of size
q
+
t
in
PG
(
2
,
q
)
such that every line contains 0, 2 or
t
of its points. We use field reduction to give a different point of view on the class of
translation
arcs. Starting from a particular
F
2
-linear set, called an
i
-club
, we reconstruct the
projective triads
, the
translation hyperovals
as well as the translation arcs constructed by Korchmáros-Mazzocca, Gács-Weiner and Limbupasiriporn. We show the KM-arcs of type
q
/
4
recently constructed by Vandendriessche are translation arcs and fit in this family. Finally, we construct a family of KM-arcs of type
q
/
4
. We show that this family, apart from new examples that are not translation KM-arcs, contains all translation KM-arcs of type
q
/
4
.</description><identifier>ISSN: 0925-9899</identifier><identifier>EISSN: 1572-9192</identifier><identifier>DOI: 10.1007/s10801-015-0661-7</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Combinatorics ; Computer Science ; Convex and Discrete Geometry ; Group Theory and Generalizations ; Lattices ; Mathematics ; Mathematics and Statistics ; Order ; Ordered Algebraic Structures</subject><ispartof>Journal of algebraic combinatorics, 2016-08, Vol.44 (1), p.131-164</ispartof><rights>Springer Science+Business Media New York 2016</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c288t-37374eb733dcde11d24e2d8ee01807f7e58241a0057c51d3e807e9b219a9849f3</citedby><cites>FETCH-LOGICAL-c288t-37374eb733dcde11d24e2d8ee01807f7e58241a0057c51d3e807e9b219a9849f3</cites><orcidid>0000-0002-4957-6911</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,780,784,27924,27925</link.rule.ids></links><search><creatorcontrib>De Boeck, Maarten</creatorcontrib><creatorcontrib>Van de Voorde, Geertrui</creatorcontrib><title>A linear set view on KM-arcs</title><title>Journal of algebraic combinatorics</title><addtitle>J Algebr Comb</addtitle><description>In this paper, we study KM-arcs of type
t
, i.e., point sets of size
q
+
t
in
PG
(
2
,
q
)
such that every line contains 0, 2 or
t
of its points. We use field reduction to give a different point of view on the class of
translation
arcs. Starting from a particular
F
2
-linear set, called an
i
-club
, we reconstruct the
projective triads
, the
translation hyperovals
as well as the translation arcs constructed by Korchmáros-Mazzocca, Gács-Weiner and Limbupasiriporn. We show the KM-arcs of type
q
/
4
recently constructed by Vandendriessche are translation arcs and fit in this family. Finally, we construct a family of KM-arcs of type
q
/
4
. We show that this family, apart from new examples that are not translation KM-arcs, contains all translation KM-arcs of type
q
/
4
.</description><subject>Combinatorics</subject><subject>Computer Science</subject><subject>Convex and Discrete Geometry</subject><subject>Group Theory and Generalizations</subject><subject>Lattices</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Order</subject><subject>Ordered Algebraic Structures</subject><issn>0925-9899</issn><issn>1572-9192</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2016</creationdate><recordtype>article</recordtype><recordid>eNp9j8FKAzEQhoMouFYfQPCwLxCdSTZN5liKWrHiRc8hzc7KlrorSVV8e1PWs6eBn_l-_k-IS4RrBLA3GcEBSkAjYT5HaY9EhcYqSUjqWFRAykhyRKfiLOctAJBDU4mrRb3rBw6pzryvv3r-rsehfnySIcV8Lk66sMt88Xdn4vXu9mW5kuvn-4flYi2jcm4vtdW24Y3Vuo0tI7aqYdU6ZkAHtrNsnGowABgbDbaaS8q0UUiBXEOdngmcemMac07c-Y_Uv4f04xH8Qc9Per7o-YOet4VRE5PL7_DGyW_HzzSUmf9Av0JUT6M</recordid><startdate>20160801</startdate><enddate>20160801</enddate><creator>De Boeck, Maarten</creator><creator>Van de Voorde, Geertrui</creator><general>Springer US</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-4957-6911</orcidid></search><sort><creationdate>20160801</creationdate><title>A linear set view on KM-arcs</title><author>De Boeck, Maarten ; Van de Voorde, Geertrui</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c288t-37374eb733dcde11d24e2d8ee01807f7e58241a0057c51d3e807e9b219a9849f3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2016</creationdate><topic>Combinatorics</topic><topic>Computer Science</topic><topic>Convex and Discrete Geometry</topic><topic>Group Theory and Generalizations</topic><topic>Lattices</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Order</topic><topic>Ordered Algebraic Structures</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>De Boeck, Maarten</creatorcontrib><creatorcontrib>Van de Voorde, Geertrui</creatorcontrib><collection>CrossRef</collection><jtitle>Journal of algebraic combinatorics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>De Boeck, Maarten</au><au>Van de Voorde, Geertrui</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>A linear set view on KM-arcs</atitle><jtitle>Journal of algebraic combinatorics</jtitle><stitle>J Algebr Comb</stitle><date>2016-08-01</date><risdate>2016</risdate><volume>44</volume><issue>1</issue><spage>131</spage><epage>164</epage><pages>131-164</pages><issn>0925-9899</issn><eissn>1572-9192</eissn><abstract>In this paper, we study KM-arcs of type
t
, i.e., point sets of size
q
+
t
in
PG
(
2
,
q
)
such that every line contains 0, 2 or
t
of its points. We use field reduction to give a different point of view on the class of
translation
arcs. Starting from a particular
F
2
-linear set, called an
i
-club
, we reconstruct the
projective triads
, the
translation hyperovals
as well as the translation arcs constructed by Korchmáros-Mazzocca, Gács-Weiner and Limbupasiriporn. We show the KM-arcs of type
q
/
4
recently constructed by Vandendriessche are translation arcs and fit in this family. Finally, we construct a family of KM-arcs of type
q
/
4
. We show that this family, apart from new examples that are not translation KM-arcs, contains all translation KM-arcs of type
q
/
4
.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s10801-015-0661-7</doi><tpages>34</tpages><orcidid>https://orcid.org/0000-0002-4957-6911</orcidid></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0925-9899 |
| ispartof | Journal of algebraic combinatorics, 2016-08, Vol.44 (1), p.131-164 |
| issn | 0925-9899 1572-9192 |
| language | eng |
| recordid | cdi_crossref_primary_10_1007_s10801_015_0661_7 |
| source | Springer Nature |
| subjects | Combinatorics Computer Science Convex and Discrete Geometry Group Theory and Generalizations Lattices Mathematics Mathematics and Statistics Order Ordered Algebraic Structures |
| title | A linear set view on KM-arcs |
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