Open AccessNonexistence of some ternary linear codes with minimum weight -2 modulo 9
Author(s) -
Toshiharu Sawashima,
Tatsuya Maruta
Publication year - 2023
Publication title -
advances in mathematics of communications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.601
H-Index - 26
eISSN - 1930-5346
pISSN - 1930-5338
DOI - 10.3934/amc.2021052
Subject(s) - mathematics , modulo , combinatorics , dimension (graph theory) , discrete mathematics , algebra over a field , pure mathematics
One of the fundamental problems in coding theory is to find \begin{document}$ n_q(k,d) $\end{document} , the minimum length \begin{document}$ n $\end{document} for which a linear code of length \begin{document}$ n $\end{document} , dimension \begin{document}$ k $\end{document} , and the minimum weight \begin{document}$ d $\end{document} over the field of order \begin{document}$ q $\end{document} exists. The problem of determining the values of \begin{document}$ n_q(k,d) $\end{document} is known as the optimal linear codes problem. Using the geometric methods through projective geometry and a new extension theorem given by Kanda (2020), we determine \begin{document}$ n_3(6,d) $\end{document} for some values of \begin{document}$ d $\end{document} by proving the nonexistence of linear codes with certain parameters.