Counting the number of non-isotopic Taniguchi semifields
We investigate the isotopy question for Taniguchi semifields. We give a complete characterization when two Taniguchi semifields are isotopic. We further give precise upper and lower bounds for the total number of non-isotopic Taniguchi semifields, proving that there are around p m + s non-isotopic T...
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| Veröffentlicht in: | Designs, codes, and cryptography codes, and cryptography, 2024-03, Vol.92 (3), p.681-694 |
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| container_title | Designs, codes, and cryptography |
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| creator | Göloğlu, Faruk Kölsch, Lukas |
| description | We investigate the isotopy question for Taniguchi semifields. We give a complete characterization when two Taniguchi semifields are isotopic. We further give precise upper and lower bounds for the total number of non-isotopic Taniguchi semifields, proving that there are around
p
m
+
s
non-isotopic Taniguchi semifields of order
p
2
m
where
s
is the largest divisor of
m
with
2
s
≠
m
. This result proves that the family of Taniguchi semifields is (asymptotically) the largest known family of semifields of odd order. The key ingredient of the proofs is a technique to determine isotopy that uses group theory to exploit the existence of certain large subgroups of the autotopism group of a semifield. |
| doi_str_mv | 10.1007/s10623-023-01262-0 |
| format | Article |
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p
m
+
s
non-isotopic Taniguchi semifields of order
p
2
m
where
s
is the largest divisor of
m
with
2
s
≠
m
. This result proves that the family of Taniguchi semifields is (asymptotically) the largest known family of semifields of odd order. The key ingredient of the proofs is a technique to determine isotopy that uses group theory to exploit the existence of certain large subgroups of the autotopism group of a semifield.</description><identifier>ISSN: 0925-1022</identifier><identifier>EISSN: 1573-7586</identifier><identifier>DOI: 10.1007/s10623-023-01262-0</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Coding and Cryptography 2022 ; Coding and Information Theory ; Computer Science ; Cryptology ; Discrete Mathematics in Computer Science ; Group theory ; Lower bounds ; Subgroups</subject><ispartof>Designs, codes, and cryptography, 2024-03, Vol.92 (3), p.681-694</ispartof><rights>The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c314t-4d03151485d5777d7ef69bd3d621d2d3b2cd20a7ebce709ffef5937a9f75a0a83</cites><orcidid>0000-0002-2966-0710</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s10623-023-01262-0$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10623-023-01262-0$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Göloğlu, Faruk</creatorcontrib><creatorcontrib>Kölsch, Lukas</creatorcontrib><title>Counting the number of non-isotopic Taniguchi semifields</title><title>Designs, codes, and cryptography</title><addtitle>Des. Codes Cryptogr</addtitle><description>We investigate the isotopy question for Taniguchi semifields. We give a complete characterization when two Taniguchi semifields are isotopic. We further give precise upper and lower bounds for the total number of non-isotopic Taniguchi semifields, proving that there are around
p
m
+
s
non-isotopic Taniguchi semifields of order
p
2
m
where
s
is the largest divisor of
m
with
2
s
≠
m
. This result proves that the family of Taniguchi semifields is (asymptotically) the largest known family of semifields of odd order. The key ingredient of the proofs is a technique to determine isotopy that uses group theory to exploit the existence of certain large subgroups of the autotopism group of a semifield.</description><subject>Coding and Cryptography 2022</subject><subject>Coding and Information Theory</subject><subject>Computer Science</subject><subject>Cryptology</subject><subject>Discrete Mathematics in Computer Science</subject><subject>Group theory</subject><subject>Lower bounds</subject><subject>Subgroups</subject><issn>0925-1022</issn><issn>1573-7586</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><recordid>eNp9kLFOwzAURS0EEqXwA0yRmA3v2XGcjKgCilSJpcyWE9utq9YudjLw9yQKEhvD1VvOuU-6hNwjPCKAfMoIFeMUpiCrGIULskAhOZWiri7JAhomKAJj1-Qm5wMAIAe2IPUqDqH3YVf0e1uE4dTaVERXhBioz7GPZ98VWx38buj2vsj25J23R5NvyZXTx2zvfu-SfL6-bFdruvl4e189b2jHsexpaYCjwLIWRkgpjbSualrDTcXQMMNb1hkGWtq2sxIa56wTDZe6cVJo0DVfkoe595zi12Bzrw5xSGF8qTigKEtWShwpNlNdijkn69Q5-ZNO3wpBTQupeSEFU6aFFIwSn6U8wmFn01_1P9YP9hJoPQ</recordid><startdate>20240301</startdate><enddate>20240301</enddate><creator>Göloğlu, Faruk</creator><creator>Kölsch, Lukas</creator><general>Springer US</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-2966-0710</orcidid></search><sort><creationdate>20240301</creationdate><title>Counting the number of non-isotopic Taniguchi semifields</title><author>Göloğlu, Faruk ; Kölsch, Lukas</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c314t-4d03151485d5777d7ef69bd3d621d2d3b2cd20a7ebce709ffef5937a9f75a0a83</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Coding and Cryptography 2022</topic><topic>Coding and Information Theory</topic><topic>Computer Science</topic><topic>Cryptology</topic><topic>Discrete Mathematics in Computer Science</topic><topic>Group theory</topic><topic>Lower bounds</topic><topic>Subgroups</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Göloğlu, Faruk</creatorcontrib><creatorcontrib>Kölsch, Lukas</creatorcontrib><collection>CrossRef</collection><jtitle>Designs, codes, and cryptography</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Göloğlu, Faruk</au><au>Kölsch, Lukas</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Counting the number of non-isotopic Taniguchi semifields</atitle><jtitle>Designs, codes, and cryptography</jtitle><stitle>Des. Codes Cryptogr</stitle><date>2024-03-01</date><risdate>2024</risdate><volume>92</volume><issue>3</issue><spage>681</spage><epage>694</epage><pages>681-694</pages><issn>0925-1022</issn><eissn>1573-7586</eissn><abstract>We investigate the isotopy question for Taniguchi semifields. We give a complete characterization when two Taniguchi semifields are isotopic. We further give precise upper and lower bounds for the total number of non-isotopic Taniguchi semifields, proving that there are around
p
m
+
s
non-isotopic Taniguchi semifields of order
p
2
m
where
s
is the largest divisor of
m
with
2
s
≠
m
. This result proves that the family of Taniguchi semifields is (asymptotically) the largest known family of semifields of odd order. The key ingredient of the proofs is a technique to determine isotopy that uses group theory to exploit the existence of certain large subgroups of the autotopism group of a semifield.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s10623-023-01262-0</doi><tpages>14</tpages><orcidid>https://orcid.org/0000-0002-2966-0710</orcidid><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
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| ispartof | Designs, codes, and cryptography, 2024-03, Vol.92 (3), p.681-694 |
| issn | 0925-1022 1573-7586 |
| language | eng |
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| source | Springer Nature - Complete Springer Journals |
| subjects | Coding and Cryptography 2022 Coding and Information Theory Computer Science Cryptology Discrete Mathematics in Computer Science Group theory Lower bounds Subgroups |
| title | Counting the number of non-isotopic Taniguchi semifields |
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