Arithmetic completely regular codes
In this paper, we explore completely regular codes in the Hamming graphs and related graphs. Experimental evidence suggests that many completely regular codes have the property that the eigenvalues of the code are in arithmetic progression. In order to better understand these "arithmetic comple...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2016-02-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
| Subjects: | |
| Online Access: | https://dmtcs.episciences.org/2150/pdf |
| _version_ | 1797270085186879488 |
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| author | Jacobus Koolen Woo Sun Lee William Martin Hajime Tanaka |
| author_facet | Jacobus Koolen Woo Sun Lee William Martin Hajime Tanaka |
| author_sort | Jacobus Koolen |
| collection | DOAJ |
| description | In this paper, we explore completely regular codes in the Hamming graphs and related graphs. Experimental evidence suggests that many completely regular codes have the property that the eigenvalues of the code are in arithmetic progression. In order to better understand these "arithmetic completely regular codes", we focus on cartesian products of completely regular codes and products of their corresponding coset graphs in the additive case. Employing earlier results, we are then able to prove a theorem which nearly classifies these codes in the case where the graph admits a completely regular partition into such codes (e.g, the cosets of some additive completely regular code). Connections to the theory of distance-regular graphs are explored and several open questions are posed. |
| first_indexed | 2024-04-25T01:58:39Z |
| format | Article |
| id | doaj.art-881bcad9bb444d79a6cbf7b4bff41bb7 |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T01:58:39Z |
| publishDate | 2016-02-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-881bcad9bb444d79a6cbf7b4bff41bb72024-03-07T15:29:03ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502016-02-01Vol. 17 no. 3PRIMA 201310.46298/dmtcs.21502150Arithmetic completely regular codesJacobus Koolen0Woo Sun Lee1William Martin2Hajime Tanaka3Wu Wen-Tsun Key Laboratory of Mathematics, Chinese Academy of SciencesDepartment of Mathematics, POSTECHDepartment of Mathematics [Worcester]Research Center for Pure and Applied Mathematics, GSIS, Tohoku UniversityIn this paper, we explore completely regular codes in the Hamming graphs and related graphs. Experimental evidence suggests that many completely regular codes have the property that the eigenvalues of the code are in arithmetic progression. In order to better understand these "arithmetic completely regular codes", we focus on cartesian products of completely regular codes and products of their corresponding coset graphs in the additive case. Employing earlier results, we are then able to prove a theorem which nearly classifies these codes in the case where the graph admits a completely regular partition into such codes (e.g, the cosets of some additive completely regular code). Connections to the theory of distance-regular graphs are explored and several open questions are posed.https://dmtcs.episciences.org/2150/pdfleonard’s theoremcompletely regular codedistance-regular graphhamming graphcoset graph[info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| spellingShingle | Jacobus Koolen Woo Sun Lee William Martin Hajime Tanaka Arithmetic completely regular codes Discrete Mathematics & Theoretical Computer Science leonard’s theorem completely regular code distance-regular graph hamming graph coset graph [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| title | Arithmetic completely regular codes |
| title_full | Arithmetic completely regular codes |
| title_fullStr | Arithmetic completely regular codes |
| title_full_unstemmed | Arithmetic completely regular codes |
| title_short | Arithmetic completely regular codes |
| title_sort | arithmetic completely regular codes |
| topic | leonard’s theorem completely regular code distance-regular graph hamming graph coset graph [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] |
| url | https://dmtcs.episciences.org/2150/pdf |
| work_keys_str_mv | AT jacobuskoolen arithmeticcompletelyregularcodes AT woosunlee arithmeticcompletelyregularcodes AT williammartin arithmeticcompletelyregularcodes AT hajimetanaka arithmeticcompletelyregularcodes |