A note on small weight codewords of projective geometric codes and on the smallest sets of even type
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2023
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866908789033140224 |
|---|---|
| author | Adriaensen, Sam |
| author_facet | Adriaensen, Sam |
| contents | In this paper, we study the codes $\mathcal C_k(n,q)$ arising from the incidence of points and $k$-spaces in $\text{PG}(n,q)$ over the field $\mathbb F_p$, with $q = p^h$, $p$ prime. We classify all codewords of minimum weight of the dual code $\mathcal C_k(n,q)^\perp$ in case $q \in \{4,8\}$. This is equivalent to classifying the smallest sets of even type in $\text{PG}(n,q)$ for $q \in \{4,8\}$. We also provide shorter proofs for some already known results, namely of the best known lower bound on the minimum weight of $\mathcal C_k(n,q)^\perp$ for general values of $q$, and of the classification of all codewords of $\mathcal C_{n-1}(n,q)$ of weight up to $2q^{n-1}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_04718 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A note on small weight codewords of projective geometric codes and on the smallest sets of even type Adriaensen, Sam Combinatorics Information Theory 51E20, 05B25, 94B05 In this paper, we study the codes $\mathcal C_k(n,q)$ arising from the incidence of points and $k$-spaces in $\text{PG}(n,q)$ over the field $\mathbb F_p$, with $q = p^h$, $p$ prime. We classify all codewords of minimum weight of the dual code $\mathcal C_k(n,q)^\perp$ in case $q \in \{4,8\}$. This is equivalent to classifying the smallest sets of even type in $\text{PG}(n,q)$ for $q \in \{4,8\}$. We also provide shorter proofs for some already known results, namely of the best known lower bound on the minimum weight of $\mathcal C_k(n,q)^\perp$ for general values of $q$, and of the classification of all codewords of $\mathcal C_{n-1}(n,q)$ of weight up to $2q^{n-1}$. |
| title | A note on small weight codewords of projective geometric codes and on the smallest sets of even type |
| topic | Combinatorics Information Theory 51E20, 05B25, 94B05 |
| url | https://arxiv.org/abs/2302.04718 |