A note on small weight codewords of projective geometric codes and on the smallest sets of even type

TL;DR

This paper classifies small weight codewords in certain projective geometric codes over fields with four and eight elements, providing new insights and simplified proofs for known bounds and classifications.

Contribution

It offers a complete classification of minimum weight codewords in dual codes for specific small fields and simplifies existing proofs of bounds and classifications in projective geometric codes.

Findings

Classified all minimum weight codewords for q=4,8.

Provided shorter proofs for known lower bounds.

Extended classification results to specific cases.

Abstract

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 $2q^{n-1}$.