Conway Groupoids and Completely Transitive Codes
To each supersimple 2-( n ,4,λ) design D one associates a ‘Conway groupoid’, which may be thought of as a natural generalisation of Conway’s Mathieu groupoid M 13 which is constructed from P 3 . We show that Sp 2 m (2) and 2 2 m . Sp 2 m (2) naturally occur as Conway groupoids associated to certain...
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| Veröffentlicht in: | Combinatorica (Budapest. 1981) 2018-04, Vol.38 (2), p.399-442 |
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| container_title | Combinatorica (Budapest. 1981) |
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| creator | Gill, Nick Gillespie, Neil I. Semeraro, Jason |
| description | To each supersimple 2-(
n
,4,λ) design
D
one associates a ‘Conway groupoid’, which may be thought of as a natural generalisation of Conway’s Mathieu groupoid
M
13
which is constructed from P
3
.
We show that Sp
2
m
(2) and 2
2
m
. Sp
2
m
(2) naturally occur as Conway groupoids associated to certain designs. It is shown that the incidence matrix associated to one of these designs generates a new family of completely transitive F
2
-linear codes with minimum distance 4 and covering radius 3, whereas the incidence matrix of the other design gives an alternative construction of a previously known family of completely transitive codes.
We also give a new characterization of
M
13
and prove that, for a fixed λ > 0; there are finitely many Conway groupoids for which the set of morphisms does not contain all elements of the full alternating group. |
| doi_str_mv | 10.1007/s00493-016-3433-7 |
| format | Article |
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n
,4,λ) design
D
one associates a ‘Conway groupoid’, which may be thought of as a natural generalisation of Conway’s Mathieu groupoid
M
13
which is constructed from P
3
.
We show that Sp
2
m
(2) and 2
2
m
. Sp
2
m
(2) naturally occur as Conway groupoids associated to certain designs. It is shown that the incidence matrix associated to one of these designs generates a new family of completely transitive F
2
-linear codes with minimum distance 4 and covering radius 3, whereas the incidence matrix of the other design gives an alternative construction of a previously known family of completely transitive codes.
We also give a new characterization of
M
13
and prove that, for a fixed λ > 0; there are finitely many Conway groupoids for which the set of morphisms does not contain all elements of the full alternating group.</description><identifier>ISSN: 0209-9683</identifier><identifier>EISSN: 1439-6912</identifier><identifier>DOI: 10.1007/s00493-016-3433-7</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Combinatorics ; Incidence ; Mathematics ; Mathematics and Statistics ; Original Paper</subject><ispartof>Combinatorica (Budapest. 1981), 2018-04, Vol.38 (2), p.399-442</ispartof><rights>János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2017</rights><rights>COPYRIGHT 2018 Springer</rights><rights>Copyright Springer Science & Business Media 2018</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c307t-8888e855d71b41922e926461dce393f02f045991ab38ce8b70f721199caea9a03</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00493-016-3433-7$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00493-016-3433-7$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Gill, Nick</creatorcontrib><creatorcontrib>Gillespie, Neil I.</creatorcontrib><creatorcontrib>Semeraro, Jason</creatorcontrib><title>Conway Groupoids and Completely Transitive Codes</title><title>Combinatorica (Budapest. 1981)</title><addtitle>Combinatorica</addtitle><description>To each supersimple 2-(
n
,4,λ) design
D
one associates a ‘Conway groupoid’, which may be thought of as a natural generalisation of Conway’s Mathieu groupoid
M
13
which is constructed from P
3
.
We show that Sp
2
m
(2) and 2
2
m
. Sp
2
m
(2) naturally occur as Conway groupoids associated to certain designs. It is shown that the incidence matrix associated to one of these designs generates a new family of completely transitive F
2
-linear codes with minimum distance 4 and covering radius 3, whereas the incidence matrix of the other design gives an alternative construction of a previously known family of completely transitive codes.
We also give a new characterization of
M
13
and prove that, for a fixed λ > 0; there are finitely many Conway groupoids for which the set of morphisms does not contain all elements of the full alternating group.</description><subject>Combinatorics</subject><subject>Incidence</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Original Paper</subject><issn>0209-9683</issn><issn>1439-6912</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2018</creationdate><recordtype>article</recordtype><recordid>eNp1kE9rwzAMxc3YYF23D7BbYWd3kp3E9rGE_YPCLt3ZuIlSUtI4s9ONfvu5ZLDTpIPg8X6SeIzdIywRQD1GgMxIDlhwmUnJ1QWbYSYNLwyKSzYDAYabQstrdhPjHgC0xHzGoPT9tzstXoI_Dr6t48L19aL0h6GjkbrTYhNcH9ux_aKk1hRv2VXjukh3v3POPp6fNuUrX7-_vJWrNa8kqJHrVKTzvFa4zdAIQUYUWYF1RdLIBkQDWW4Muq3UFemtgkYJRGMqR844kHP2MO0dgv88Uhzt3h9Dn05aAVKDhEJjci0n1851ZNu-8WNwVeqaDm3le2rapK-UMAjaCJMAnIAq-BgDNXYI7cGFk0Ww5yTtlKRNSdpzklYlRkxMTN5-R-Hvlf-hH5U2c5M</recordid><startdate>20180401</startdate><enddate>20180401</enddate><creator>Gill, Nick</creator><creator>Gillespie, Neil I.</creator><creator>Semeraro, Jason</creator><general>Springer Berlin Heidelberg</general><general>Springer</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20180401</creationdate><title>Conway Groupoids and Completely Transitive Codes</title><author>Gill, Nick ; Gillespie, Neil I. ; Semeraro, Jason</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c307t-8888e855d71b41922e926461dce393f02f045991ab38ce8b70f721199caea9a03</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2018</creationdate><topic>Combinatorics</topic><topic>Incidence</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Original Paper</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Gill, Nick</creatorcontrib><creatorcontrib>Gillespie, Neil I.</creatorcontrib><creatorcontrib>Semeraro, Jason</creatorcontrib><collection>CrossRef</collection><jtitle>Combinatorica (Budapest. 1981)</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Gill, Nick</au><au>Gillespie, Neil I.</au><au>Semeraro, Jason</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Conway Groupoids and Completely Transitive Codes</atitle><jtitle>Combinatorica (Budapest. 1981)</jtitle><stitle>Combinatorica</stitle><date>2018-04-01</date><risdate>2018</risdate><volume>38</volume><issue>2</issue><spage>399</spage><epage>442</epage><pages>399-442</pages><issn>0209-9683</issn><eissn>1439-6912</eissn><abstract>To each supersimple 2-(
n
,4,λ) design
D
one associates a ‘Conway groupoid’, which may be thought of as a natural generalisation of Conway’s Mathieu groupoid
M
13
which is constructed from P
3
.
We show that Sp
2
m
(2) and 2
2
m
. Sp
2
m
(2) naturally occur as Conway groupoids associated to certain designs. It is shown that the incidence matrix associated to one of these designs generates a new family of completely transitive F
2
-linear codes with minimum distance 4 and covering radius 3, whereas the incidence matrix of the other design gives an alternative construction of a previously known family of completely transitive codes.
We also give a new characterization of
M
13
and prove that, for a fixed λ > 0; there are finitely many Conway groupoids for which the set of morphisms does not contain all elements of the full alternating group.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00493-016-3433-7</doi><tpages>44</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0209-9683 |
| ispartof | Combinatorica (Budapest. 1981), 2018-04, Vol.38 (2), p.399-442 |
| issn | 0209-9683 1439-6912 |
| language | eng |
| recordid | cdi_proquest_journals_2038030681 |
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| subjects | Combinatorics Incidence Mathematics Mathematics and Statistics Original Paper |
| title | Conway Groupoids and Completely Transitive Codes |
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