On the (29,5)-Arcs in <i>PG</i>(2,7) and Some Generalized Arcs in <i>PG</i>(2,<i>q</i>)
Using an exhaustive computer search, we prove that the number of inequivalent <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mn>29</mn> <mo>,</mo> <mn>5</mn> <mo>)</mo> </mrow>...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-03-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/3/320 |
| _version_ | 1818537371063287808 |
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| author | Iliya Bouyukliev Eun Ju Cheon Tatsuya Maruta Tsukasa Okazaki |
| author_facet | Iliya Bouyukliev Eun Ju Cheon Tatsuya Maruta Tsukasa Okazaki |
| author_sort | Iliya Bouyukliev |
| collection | DOAJ |
| description | Using an exhaustive computer search, we prove that the number of inequivalent <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mn>29</mn> <mo>,</mo> <mn>5</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-arcs in PG<inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mn>7</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> is exactly 22. This generalizes a result of Barlotti (see Barlotti, A. Some Topics in Finite Geometrical Structures, 1965), who constructed the first such arc from a conic. Our classification result is based on the fact that arcs and linear codes are related, which enables us to apply an algorithm for classifying the associated linear codes instead. Related to this result, several infinite families of arcs and multiple blocking sets are constructed. Lastly, the relationship between these arcs and the Barlotti arc is explored using a construction that we call transitioning. |
| first_indexed | 2024-12-11T18:49:55Z |
| format | Article |
| id | doaj.art-db3ab1086b4343baaff35065566e58e5 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-12-11T18:49:55Z |
| publishDate | 2020-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-db3ab1086b4343baaff35065566e58e52022-12-22T00:54:20ZengMDPI AGMathematics2227-73902020-03-018332010.3390/math8030320math8030320On the (29,5)-Arcs in <i>PG</i>(2,7) and Some Generalized Arcs in <i>PG</i>(2,<i>q</i>)Iliya Bouyukliev0Eun Ju Cheon1Tatsuya Maruta2Tsukasa Okazaki3Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 5000 Veliko Tarnovo, BulgariaDepartment of Mathematics and RINS, Gyeongsang National University, Jinju 52828, KoreaDepartment of Mathematical Sciences, Osaka Prefecture University, Sakai, Osaka 599-8531, JapanDepartment of Mathematical Sciences, Osaka Prefecture University, Sakai, Osaka 599-8531, JapanUsing an exhaustive computer search, we prove that the number of inequivalent <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mn>29</mn> <mo>,</mo> <mn>5</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-arcs in PG<inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mn>7</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> is exactly 22. This generalizes a result of Barlotti (see Barlotti, A. Some Topics in Finite Geometrical Structures, 1965), who constructed the first such arc from a conic. Our classification result is based on the fact that arcs and linear codes are related, which enables us to apply an algorithm for classifying the associated linear codes instead. Related to this result, several infinite families of arcs and multiple blocking sets are constructed. Lastly, the relationship between these arcs and the Barlotti arc is explored using a construction that we call transitioning.https://www.mdpi.com/2227-7390/8/3/320projective planearcblocking setlinear code, griesmer code |
| spellingShingle | Iliya Bouyukliev Eun Ju Cheon Tatsuya Maruta Tsukasa Okazaki On the (29,5)-Arcs in <i>PG</i>(2,7) and Some Generalized Arcs in <i>PG</i>(2,<i>q</i>) Mathematics projective plane arc blocking set linear code, griesmer code |
| title | On the (29,5)-Arcs in <i>PG</i>(2,7) and Some Generalized Arcs in <i>PG</i>(2,<i>q</i>) |
| title_full | On the (29,5)-Arcs in <i>PG</i>(2,7) and Some Generalized Arcs in <i>PG</i>(2,<i>q</i>) |
| title_fullStr | On the (29,5)-Arcs in <i>PG</i>(2,7) and Some Generalized Arcs in <i>PG</i>(2,<i>q</i>) |
| title_full_unstemmed | On the (29,5)-Arcs in <i>PG</i>(2,7) and Some Generalized Arcs in <i>PG</i>(2,<i>q</i>) |
| title_short | On the (29,5)-Arcs in <i>PG</i>(2,7) and Some Generalized Arcs in <i>PG</i>(2,<i>q</i>) |
| title_sort | on the 29 5 arcs in i pg i 2 7 and some generalized arcs in i pg i 2 i q i |
| topic | projective plane arc blocking set linear code, griesmer code |
| url | https://www.mdpi.com/2227-7390/8/3/320 |
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