Classification of $ \mathbf{(3 \!\mod 5)} $ arcs in $ \mathbf{ \operatorname{PG}(3,5)} $

Abstract

<p style='text-indent:20px;'>The proof of the non-existence of Griesmer <inline-formula><tex-math id="M3">\begin{document}$ [104, 4, 82]_5 $\end{document}</tex-math></inline-formula>-codes is just one of many examples where extendability results are used. In a series of papers Landjev and Rousseva have introduced the concept of <inline-formula><tex-math id="M4">\begin{document}$ (t\mod q) $\end{document}</tex-math></inline-formula>-arcs as a general framework for extendability results for codes and arcs. Here we complete the known partial classification of <inline-formula><tex-math id="M5">\begin{document}$ (3 \mod 5) $\end{document}</tex-math></inline-formula>-arcs in <inline-formula><tex-math id="M6">\begin{document}$ \operatorname{PG}(3,5) $\end{document}</tex-math></inline-formula> and uncover two missing, rather exceptional, examples disproving a conjecture of Landjev and Rousseva. As also the original non-existence proof of Griesmer <inline-formula><tex-math id="M7">\begin{document}$ [104, 4, 82]_5 $\end{document}</tex-math></inline-formula>-codes is affected, we present an extended proof to fill this gap.</p>