On additive MDS codes over small fields
| Autor: | Michel Lavrauw, Simeon Ball, Guillermo Gamboa |
|---|---|
| Přispěvatelé: | Universitat Politècnica de Catalunya. Departament de Matemàtiques, Universitat Politècnica de Catalunya. GAPCOMB - Geometric, Algebraic and Probabilistic Combinatorics |
| Jazyk: | angličtina |
| Rok vydání: | 2022 |
| Předmět: |
FOS: Computer and information sciences
Code (set theory) 94B27 51E22 Computer Networks and Communications stabiliser codes Computer Science - Information Theory 94 Information And Communication Circuits::94B Theory of error-correcting codes and error-detecting codes [Classificació AMS] Geometry quantum codes 02 engineering and technology 01 natural sciences Microbiology Combinatorics arcs FOS: Mathematics 0202 electrical engineering electronic engineering information engineering Mathematics - Combinatorics Discrete Mathematics and Combinatorics 0101 mathematics Mathematics Error-correcting codes (Information theory) Codis de correcció d'errors (Teoria de la informació) Algebra and Number Theory Conjecture Information Theory (cs.IT) Applied Mathematics 010102 general mathematics 020206 networking & telecommunications additive codes MDS conjecture 51 Geometry::51E Finite geometry and special incidence structures [Classificació AMS] MDS codes Combinatorics (math.CO) Geometria finita Matemàtiques i estadística::Geometria [Àrees temàtiques de la UPC] |
| Popis: | Let \begin{document}$ C $\end{document} be a \begin{document}$ (n,q^{2k},n-k+1)_{q^2} $\end{document} additive MDS code which is linear over \begin{document}$ {\mathbb F}_q $\end{document} . We prove that if \begin{document}$ n \geq q+k $\end{document} and \begin{document}$ k+1 $\end{document} of the projections of \begin{document}$ C $\end{document} are linear over \begin{document}$ {\mathbb F}_{q^2} $\end{document} then \begin{document}$ C $\end{document} is linear over \begin{document}$ {\mathbb F}_{q^2} $\end{document} . We use this geometrical theorem, other geometric arguments and some computations to classify all additive MDS codes over \begin{document}$ {\mathbb F}_q $\end{document} for \begin{document}$ q \in \{4,8,9\} $\end{document} . We also classify the longest additive MDS codes over \begin{document}$ {\mathbb F}_{16} $\end{document} which are linear over \begin{document}$ {\mathbb F}_4 $\end{document} . In these cases, the classifications not only verify the MDS conjecture for additive codes, but also confirm there are no additive non-linear MDS codes which perform as well as their linear counterparts. These results imply that the quantum MDS conjecture holds for \begin{document}$ q \in \{ 2,3\} $\end{document} . |
| Databáze: | OpenAIRE |
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