A construction of $ \mathbb{F}_2 $-linear cyclic, MDS codes

In this paper we construct \begin{document}$ \mathbb{F}_2 $\end{document} -linear codes over \begin{document}$ \mathbb{F}_{2}^{b} $\end{document} with length \begin{document}$ n $\end{document} and dimension \begin{document}$ n-r $\end{document} where \begin{document}$ n = rb $\end{document} . These codes have good properties, namely cyclicity, low density parity-check matrices and maximum distance separation in some cases. For the construction, we consider an odd prime \begin{document}$ p $\end{document} , let \begin{document}$ n = p-1 $\end{document} and utilize a partition of \begin{document}$ \mathbb{Z}_n $\end{document} . Then we apply a Zech logarithm to the elements of these sets and use the results to construct an index array which represents the parity-check matrix of the code. These codes are always cyclic and the density of the parity-check and the generator matrices decreases to \begin{document}$ 0 $\end{document} as \begin{document}$ n $\end{document} grows (for a fixed \begin{document}$ r $\end{document} ). When \begin{document}$ r = 2 $\end{document} we prove that these codes are always maximum distance separable. For higher \begin{document}$ r $\end{document} some of them retain this property.