On Hamming and b-symbol distance distributions of repeated-root constacyclic codes of length $$4p^s$$ over $${\pmb {\mathbb {F}}}_{p^m}+u {\pmb {\mathbb {F}}}_{p^m}$$

Abstract

Let p be a prime such that $$p^m \equiv 1\pmod {4}$$ , and $$\mathcal{R}=\mathbb F_{p^m}+u\mathbb F_{p^m}$$ . For any non-square unit $$\lambda $$ of $$\mathcal{R}$$ , the Hamming and b-symbol distances of all $$\lambda $$ -constacyclic codes of length $$4p^s$$ over $$\mathcal{R}$$ are completely determined. As examples, several good codes with new parameters are constructed. We also identified all Maximum Distance Separable constacyclic codes of length $$4p^s$$ over $$\mathcal{R}$$ with respect to the Hamming distance as well as the b-symbol distance. Also, we got some non-trivial MDS b-symbol Type 3, $$\gamma $$ -constacyclic codes of length $$4p^s$$ codes over $$\mathcal{R}$$ with respect to b-symbol distance for $$b=4$$ .