Alphabet-affine 2-neighbour-transitive codes
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912117712486400 |
|---|---|
| author | Hawtin, Daniel R. |
| author_facet | Hawtin, Daniel R. |
| contents | A code ${\mathcal C}$ is a subset of the vertex set of a Hamming graph $H(n,q)$, and ${\mathcal C}$ is $2$-neighbour-transitive if the automorphism group $G={\rm Aut}({\mathcal C})$ acts transitively on each of the sets ${\mathcal C}$, ${\mathcal C}_1$ and ${\mathcal C}_2$, where ${\mathcal C}_1$ and ${\mathcal C}_2$ are the (non-empty) sets of vertices that are distances $1$ and $2$, respectively, (but no closer) to some element of ${\mathcal C}$.
Suppose that ${\mathcal C}$ is a $2$-neighbour-transitive code with minimum distance at least $5$. For $q=2$, all `minimal' such ${\mathcal C}$ have been classified. Moreover, it has previously been shown that a subgroup of the automorphism group of the code induces an affine $2$-transitive group action on the alphabet of the Hamming graph. The main results of this paper are to show that this affine $2$-transitive group must be a subgroup of ${\rm A}Γ{\rm L}_1(q)$ and to provide a number of infinite families of examples of such codes. These examples are described via polynomial algebras related to representations of certain classical groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_08351 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Alphabet-affine 2-neighbour-transitive codes Hawtin, Daniel R. Combinatorics 05E18 (Primary) 20B25, 94B25 (Secondary) A code ${\mathcal C}$ is a subset of the vertex set of a Hamming graph $H(n,q)$, and ${\mathcal C}$ is $2$-neighbour-transitive if the automorphism group $G={\rm Aut}({\mathcal C})$ acts transitively on each of the sets ${\mathcal C}$, ${\mathcal C}_1$ and ${\mathcal C}_2$, where ${\mathcal C}_1$ and ${\mathcal C}_2$ are the (non-empty) sets of vertices that are distances $1$ and $2$, respectively, (but no closer) to some element of ${\mathcal C}$. Suppose that ${\mathcal C}$ is a $2$-neighbour-transitive code with minimum distance at least $5$. For $q=2$, all `minimal' such ${\mathcal C}$ have been classified. Moreover, it has previously been shown that a subgroup of the automorphism group of the code induces an affine $2$-transitive group action on the alphabet of the Hamming graph. The main results of this paper are to show that this affine $2$-transitive group must be a subgroup of ${\rm A}Γ{\rm L}_1(q)$ and to provide a number of infinite families of examples of such codes. These examples are described via polynomial algebras related to representations of certain classical groups. |
| title | Alphabet-affine 2-neighbour-transitive codes |
| topic | Combinatorics 05E18 (Primary) 20B25, 94B25 (Secondary) |
| url | https://arxiv.org/abs/2411.08351 |