On extended 1-perfect bitrades
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2020
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866912005526388736 |
|---|---|
| author | Bespalov, Evgeny A. Krotov, Denis S. |
| author_facet | Bespalov, Evgeny A. Krotov, Denis S. |
| contents | Extended $1$-perfect codes in the Hamming scheme $H(n,q)$ can be equivalently defined as codes that turn to $1$-perfect codes after puncturing in any coordinate, as completely regular codes with certain intersection array, as uniformly packed codes with certain weight coefficients, as diameter perfect codes with respect to a certain anticode, as distance-$4$ codes with certain dual distances. We define extended $1$-perfect bitrades in $H(n,q)$ in five different manners, corresponding to the different definitions of extended $1$-perfect codes, and prove the equivalence of these definitions of extended $1$-perfect bitrades. For $q=2^m$, we prove that such bitrades exist if and only if $n=lq+2$. For any $q$, we prove the nonexistence of extended $1$-perfect bitrades if $n$ is odd.
Keywords: Perfect code, Extended perfect code, Bitrade, Completely regular code, Uniformly packed code. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2012_02183 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On extended 1-perfect bitrades Bespalov, Evgeny A. Krotov, Denis S. Combinatorics Discrete Mathematics 05B05, 94B25 Extended $1$-perfect codes in the Hamming scheme $H(n,q)$ can be equivalently defined as codes that turn to $1$-perfect codes after puncturing in any coordinate, as completely regular codes with certain intersection array, as uniformly packed codes with certain weight coefficients, as diameter perfect codes with respect to a certain anticode, as distance-$4$ codes with certain dual distances. We define extended $1$-perfect bitrades in $H(n,q)$ in five different manners, corresponding to the different definitions of extended $1$-perfect codes, and prove the equivalence of these definitions of extended $1$-perfect bitrades. For $q=2^m$, we prove that such bitrades exist if and only if $n=lq+2$. For any $q$, we prove the nonexistence of extended $1$-perfect bitrades if $n$ is odd. Keywords: Perfect code, Extended perfect code, Bitrade, Completely regular code, Uniformly packed code. |
| title | On extended 1-perfect bitrades |
| topic | Combinatorics Discrete Mathematics 05B05, 94B25 |
| url | https://arxiv.org/abs/2012.02183 |