Sphere packing proper colorings of an expander graph
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_ | 1866929687388749824 |
|---|---|
| author | Zhu, Honglin |
| author_facet | Zhu, Honglin |
| contents | We introduce graphical error-correcting codes, a new notion of error-correcting codes on $[q]^n$, where a code is a set of proper $q$-colorings of some fixed $n$-vertex graph $G$. We then say that a set of $M$ proper $q$-colorings of $G$ form a $(G, M, d)$ code if any pair of colorings in the set have Hamming distance at least $d$. This directly generalizes typical $(n, M, d)$ codes of $q$-ary strings of length $n$ since we can take $G$ as the empty graph on $n$ vertices.
We investigate how one-sided spectral expansion relates to the largest possible set of error-correcting colorings on a graph. For fixed $(δ, λ) \in [0, 1] \times [-1, 1]$ and positive integer $d$, let $f_{δ, λ, d}(n)$ denote the maximum $M$ such that there exists some $d$-regular graph $G$ on at most $n$ vertices with normalized second eigenvalue at most $λ$ that has a $(G, M, d)$ code. We study the growth of $f$ as $n$ goes to infinity. We partially characterize the regimes of $(δ, λ)$ where $f$ grows exponentially or is bounded by a constant, respectively. We also prove several sharp phase transitions between these regimes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_20368 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sphere packing proper colorings of an expander graph Zhu, Honglin Combinatorics Information Theory 05C15, 05C35, 05C48, 94B25, 94B65 E.4; G.2.1; G.2.2 We introduce graphical error-correcting codes, a new notion of error-correcting codes on $[q]^n$, where a code is a set of proper $q$-colorings of some fixed $n$-vertex graph $G$. We then say that a set of $M$ proper $q$-colorings of $G$ form a $(G, M, d)$ code if any pair of colorings in the set have Hamming distance at least $d$. This directly generalizes typical $(n, M, d)$ codes of $q$-ary strings of length $n$ since we can take $G$ as the empty graph on $n$ vertices. We investigate how one-sided spectral expansion relates to the largest possible set of error-correcting colorings on a graph. For fixed $(δ, λ) \in [0, 1] \times [-1, 1]$ and positive integer $d$, let $f_{δ, λ, d}(n)$ denote the maximum $M$ such that there exists some $d$-regular graph $G$ on at most $n$ vertices with normalized second eigenvalue at most $λ$ that has a $(G, M, d)$ code. We study the growth of $f$ as $n$ goes to infinity. We partially characterize the regimes of $(δ, λ)$ where $f$ grows exponentially or is bounded by a constant, respectively. We also prove several sharp phase transitions between these regimes. |
| title | Sphere packing proper colorings of an expander graph |
| topic | Combinatorics Information Theory 05C15, 05C35, 05C48, 94B25, 94B65 E.4; G.2.1; G.2.2 |
| url | https://arxiv.org/abs/2405.20368 |