Determination for minimum symbol-pair and RT weights via torsional degrees of repeated-root cyclic codes

There are various metrics for researching error-correcting codes. Especially, high-density data storage system gives the existence of inconsistency for the reading and writing process. The symbol-pair metric is motivated for outputs that have overlapping pairs of symbols in a certain channel. The Rosenbloom–Tsfasman (RT) metric is introduced since there exists a problem that is related to transmission over several parallel communication channels with some channels not available for the transmission. In this paper, we determine the minimum symbol-pair weight and RT weight of repeated-root cyclic codes over \(\mathfrak R=\mathbb {F}_{p^m}[u]/\langle u^4\rangle \) of length \(n=p^k\). For the determination, we explicitly present third torsional degree for all different types of cyclic codes over \(\mathfrak R\) of length n.