Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2020
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2012.02183 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Table of 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.