Minimal codewords arising from the incidence of points and hyperplanes in projective spaces

Abstract

<p style='text-indent:20px;'>Over the past few years, the codes <inline-formula><tex-math id="M1">\begin{document}$ {\mathcal{C}}_{n-1}(n,q) $\end{document}</tex-math></inline-formula> arising from the incidence of points and hyperplanes in the projective space <inline-formula><tex-math id="M2">\begin{document}$ {\rm{PG}}(n,q) $\end{document}</tex-math></inline-formula> attracted a lot of attention. In particular, small weight codewords of <inline-formula><tex-math id="M3">\begin{document}$ {\mathcal{C}}_{n-1}(n,q) $\end{document}</tex-math></inline-formula> are a topic of investigation. The main result of this work states that, if <inline-formula><tex-math id="M4">\begin{document}$ q $\end{document}</tex-math></inline-formula> is large enough and not prime, a codeword having weight smaller than roughly <inline-formula><tex-math id="M5">\begin{document}$ \frac{1}{2^{n-2}}q^{n-1}\sqrt{q} $\end{document}</tex-math></inline-formula> can be written as a linear combination of a few hyperplanes. Consequently, we use this result to provide a graph-theoretical sufficient condition for these codewords of small weight to be minimal.</p>

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

“…√ qq n−1 for q = p h with h > 1 in [BD23], building on results by Szőnyi and Weiner [SW18]. The classification was generalised to the codes C k (n, q) in [AD21] for codewords of weight up to roughly 3q k .…”

Small weight codewords of projective geometric codes

“…Finally, the second author and Bartoli [8] showed that if q is not prime and large enough, then codewords of C n−1 (n, q) up to weight roughly 1 2 n−2 q n−1 √ q are linear combinations of exactly wt(c) θ n−1 hyperplanes. One of the aims of this paper is to remove the exponential factor 1 2 n−2 .…”