Cyclotomic Trace Codes
A generalization of Ding’s construction is proposed that employs as a defining set the collection of the <i>s</i>th powers (<inline-formula> <math display="inline"> <semantics> <mrow> <mi>s</mi> <mo>≥</mo> <...
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| Format: | Article |
| Language: | English |
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MDPI AG
2019-08-01
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| Series: | Algorithms |
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| Online Access: | https://www.mdpi.com/1999-4893/12/8/168 |
| _version_ | 1829518146566881280 |
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| author | Dean Crnković Andrea Švob Vladimir D. Tonchev |
| author_facet | Dean Crnković Andrea Švob Vladimir D. Tonchev |
| author_sort | Dean Crnković |
| collection | DOAJ |
| description | A generalization of Ding’s construction is proposed that employs as a defining set the collection of the <i>s</i>th powers (<inline-formula> <math display="inline"> <semantics> <mrow> <mi>s</mi> <mo>≥</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula>) of all nonzero elements in <inline-formula> <math display="inline"> <semantics> <mrow> <mi>G</mi> <mi>F</mi> <mo>(</mo> <msup> <mi>p</mi> <mi>m</mi> </msup> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, where <inline-formula> <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo>≥</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula> is prime. Some of the resulting codes are optimal or near-optimal and include projective codes over <inline-formula> <math display="inline"> <semantics> <mrow> <mi>G</mi> <mi>F</mi> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> that give rise to optimal or near optimal quantum codes. In addition, the codes yield interesting combinatorial structures, such as strongly regular graphs and block designs. |
| first_indexed | 2024-12-16T14:14:35Z |
| format | Article |
| id | doaj.art-0e029396a3ed4af08f65177b02c079b4 |
| institution | Directory Open Access Journal |
| issn | 1999-4893 |
| language | English |
| last_indexed | 2024-12-16T14:14:35Z |
| publishDate | 2019-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Algorithms |
| spelling | doaj.art-0e029396a3ed4af08f65177b02c079b42022-12-21T22:28:38ZengMDPI AGAlgorithms1999-48932019-08-0112816810.3390/a12080168a12080168Cyclotomic Trace CodesDean Crnković0Andrea Švob1Vladimir D. Tonchev2Department of Mathematics, University of Rijeka, Radmile Matejčić 2, 51000 Rijeka, CroatiaDepartment of Mathematics, University of Rijeka, Radmile Matejčić 2, 51000 Rijeka, CroatiaDepartment of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931, USAA generalization of Ding’s construction is proposed that employs as a defining set the collection of the <i>s</i>th powers (<inline-formula> <math display="inline"> <semantics> <mrow> <mi>s</mi> <mo>≥</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula>) of all nonzero elements in <inline-formula> <math display="inline"> <semantics> <mrow> <mi>G</mi> <mi>F</mi> <mo>(</mo> <msup> <mi>p</mi> <mi>m</mi> </msup> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, where <inline-formula> <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo>≥</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula> is prime. Some of the resulting codes are optimal or near-optimal and include projective codes over <inline-formula> <math display="inline"> <semantics> <mrow> <mi>G</mi> <mi>F</mi> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> that give rise to optimal or near optimal quantum codes. In addition, the codes yield interesting combinatorial structures, such as strongly regular graphs and block designs.https://www.mdpi.com/1999-4893/12/8/168linear codetwo-weight codestrongly regular graphblock design |
| spellingShingle | Dean Crnković Andrea Švob Vladimir D. Tonchev Cyclotomic Trace Codes Algorithms linear code two-weight code strongly regular graph block design |
| title | Cyclotomic Trace Codes |
| title_full | Cyclotomic Trace Codes |
| title_fullStr | Cyclotomic Trace Codes |
| title_full_unstemmed | Cyclotomic Trace Codes |
| title_short | Cyclotomic Trace Codes |
| title_sort | cyclotomic trace codes |
| topic | linear code two-weight code strongly regular graph block design |
| url | https://www.mdpi.com/1999-4893/12/8/168 |
| work_keys_str_mv | AT deancrnkovic cyclotomictracecodes AT andreasvob cyclotomictracecodes AT vladimirdtonchev cyclotomictracecodes |