Constructions of Partial MDS Codes Over Small Fields

Ryan Gabrys; Eitan Yaakobi; Mario Blaum; Paul H. Siegel

doi:10.1109/tit.2018.2890201

Constructions of Partial MDS Codes Over Small Fields

IEEE Transactions on Information Theory ◽

10.1109/tit.2018.2890201 ◽

2019 ◽

Vol 65 (6) ◽

pp. 3692-3701 ◽

Cited By ~ 7

Author(s):

Ryan Gabrys ◽

Eitan Yaakobi ◽

Mario Blaum ◽

Paul H. Siegel

Keyword(s):

Mds Codes ◽

Small Fields

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Quantum MDS codes over small fields

2015 IEEE International Symposium on Information Theory (ISIT) ◽

10.1109/isit.2015.7282626 ◽

2015 ◽

Cited By ~ 15

Author(s):

Markus Grassl ◽

Martin Rotteler

Keyword(s):

Mds Codes ◽

Small Fields ◽

Quantum Mds Codes

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Constructions of partial MDS codes over small fields

2017 IEEE International Symposium on Information Theory (ISIT) ◽

10.1109/isit.2017.8006478 ◽

2017 ◽

Cited By ~ 7

Author(s):

Ryan Gabrys ◽

Eitan Yaakobi ◽

Mario Blaum ◽

Paul H. Siegel

Keyword(s):

Mds Codes ◽

Small Fields

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On additive MDS codes over small fields

Advances in Mathematics of Communications ◽

10.3934/amc.2021024 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Simeon Ball ◽

Guillermo Gamboa ◽

Michel Lavrauw

Keyword(s):

Mds Codes ◽

Mds Code ◽

Additive Codes ◽

Non Linear ◽

Small Fields ◽

Mds Conjecture

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ C $\end{document}</tex-math></inline-formula> be a <inline-formula><tex-math id="M2">\begin{document}$ (n,q^{2k},n-k+1)_{q^2} $\end{document}</tex-math></inline-formula> additive MDS code which is linear over <inline-formula><tex-math id="M3">\begin{document}$ {\mathbb F}_q $\end{document}</tex-math></inline-formula>. We prove that if <inline-formula><tex-math id="M4">\begin{document}$ n \geq q+k $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ k+1 $\end{document}</tex-math></inline-formula> of the projections of <inline-formula><tex-math id="M6">\begin{document}$ C $\end{document}</tex-math></inline-formula> are linear over <inline-formula><tex-math id="M7">\begin{document}$ {\mathbb F}_{q^2} $\end{document}</tex-math></inline-formula> then <inline-formula><tex-math id="M8">\begin{document}$ C $\end{document}</tex-math></inline-formula> is linear over <inline-formula><tex-math id="M9">\begin{document}$ {\mathbb F}_{q^2} $\end{document}</tex-math></inline-formula>. We use this geometrical theorem, other geometric arguments and some computations to classify all additive MDS codes over <inline-formula><tex-math id="M10">\begin{document}$ {\mathbb F}_q $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M11">\begin{document}$ q \in \{4,8,9\} $\end{document}</tex-math></inline-formula>. We also classify the longest additive MDS codes over <inline-formula><tex-math id="M12">\begin{document}$ {\mathbb F}_{16} $\end{document}</tex-math></inline-formula> which are linear over <inline-formula><tex-math id="M13">\begin{document}$ {\mathbb F}_4 $\end{document}</tex-math></inline-formula>. In these cases, the classifications not only verify the MDS conjecture for additive codes, but also confirm there are no additive non-linear MDS codes which perform as well as their linear counterparts. These results imply that the quantum MDS conjecture holds for <inline-formula><tex-math id="M14">\begin{document}$ q \in \{ 2,3\} $\end{document}</tex-math></inline-formula>.</p>

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