A new extension theorem for ternary linear codes and its application

<p style='text-indent:20px;'>One of the fundamental problems in coding theory is to find <inline-formula><tex-math id="M3">\begin{document}$ n_q(k,d) $\end{document}</tex-math></inline-formula>, the minimum length <inline-formula><tex-math id="M4">\begin{document}$ n $\end{document}</tex-math></inline-formula> for which a linear code of length <inline-formula><tex-math id="M5">\begin{document}$ n $\end{document}</tex-math></inline-formula>, dimension <inline-formula><tex-math id="M6">\begin{document}$ k $\end{document}</tex-math></inline-formula>, and the minimum weight <inline-formula><tex-math id="M7">\begin{document}$ d $\end{document}</tex-math></inline-formula> over the field of order <inline-formula><tex-math id="M8">\begin{document}$ q $\end{document}</tex-math></inline-formula> exists. The problem of determining the values of <inline-formula><tex-math id="M9">\begin{document}$ n_q(k,d) $\end{document}</tex-math></inline-formula> is known as the optimal linear codes problem. Using the geometric methods through projective geometry and a new extension theorem given by Kanda (2020), we determine <inline-formula><tex-math id="M10">\begin{document}$ n_3(6,d) $\end{document}</tex-math></inline-formula> for some values of <inline-formula><tex-math id="M11">\begin{document}$ d $\end{document}</tex-math></inline-formula> by proving the nonexistence of linear codes with certain parameters.</p>

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On the minimum length of ternary linear codes

Designs Codes and Cryptography ◽

10.1007/s10623-011-9593-y ◽

2011 ◽

Vol 68 (1-3) ◽

pp. 407-425 ◽

Cited By ~ 1

Author(s):

Tatsuya Maruta ◽

Yusuke Oya

Keyword(s):

Linear Codes ◽

Minimum Length ◽

Ternary Linear Codes

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Nonexistence Proofs for Five Ternary Linear Codes

Journal of Mathematics and Its Applications ◽

10.29244/jmap.1.1.35-40 ◽

2002 ◽

Vol 1 (1) ◽

pp. 35

Author(s):

S. GURITMAN

Keyword(s):

Linear Code ◽

Minimum Distance ◽

Linear Codes ◽

Ternary Linear Codes

<p>An [n,k, dh-code is a ternary linear code with length n, dimension k and minimum distance d. We prove that codes with parameters [110,6, 72h, [109,6,71h, [237,6,157b, [69,7,43h, and [120,9,75h do not exist.</p>

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