On the Hamming distances of repeated-root cyclic codes of length $5p^s$

Due to the wide applications in consumer electronics, data storage systems and communication systems, cyclic codes have been an interesting research topic in coding theory. In this paper, let $p$ be a prime with $p\ge 7$. We determine the weight distributions of all cyclic codes of length $5$ over $\f_q$ and the Hamming distances of all repeated-root cyclic codes of length $5p^s$ over $\F_q$, where $q=p^m$ and both $s$ and $m$ are positive integers. Furthermore, we find all MDS cyclic codes of length $5p^s$ and take quantum synchronizable codes from repeated-root cyclic codes of length $5p^s$.