On the existence of PD-sets: Algorithms arising from automorphism groups of codes

This paper deals with the problem of determining whether a PD-set exists for a given linear code \begin{document}$ \mathcal C $\end{document} and information set \begin{document}$ I $\end{document} . A computational approach is proposed and illustrated with two exceptional codes with automorphism groups isomorphic to the sporadic simple groups \begin{document}$ \mathrm{M}_{12} $\end{document} and \begin{document}$ \mathrm{M}_{22} $\end{document} , respectively. In both cases, the existence of a PD–set is proven. In general, the algorithm works well whenever the code \begin{document}$ \mathcal C $\end{document} has a very large automorphism group.