RT distances and Hamming distances of constacyclic codes of length $$8p^s$$ 8 p s over $${\mathbb {F}}_{p^m}+u{\mathbb {F}}_{p^m}$$ F p m + u F p m

In this paper, let $$\alpha +u\beta $$ be a unit in $${\mathbb {F}}_{p^m}+u{\mathbb {F}}_{p^m}$$ $$(u^2=0),$$ where p is an odd prime, m is a positive integer and $$\beta \ne 0$$ . With the help of decomposition of the binomial $$x^{8}- \alpha _0$$ into a product of irreducible coprime polynomials and the ring $$\frac{({\mathbb {F}}_{p^m}+u{\mathbb {F}}_{p^m})[x]}{\left\langle x^{8p^s}-(\alpha +u\beta )\right\rangle }$$ is a principal ideal ring, we give the complete description of all $$(\alpha +u\beta )$$ -constacyclic codes of length $$8p^s$$ over the finite commutative chain ring $${\mathbb {F}}_{p^m}+u{\mathbb {F}}_{p^m}$$ $$(u^2=0)$$ in terms of their generator polynomials, where $$\alpha _{0}^{p^{s}}=\alpha $$ . We also find out the number of codewords in each of these constacyclic codes. Besides illustrating our results with examples, we determine duals of constacyclic codes, and as an application, we determine the self-dual, self-orthogonal, dual-containing, and linear complimentary-dual $$(\alpha +u\beta )$$ -constacyclic codes of length $$8p^s$$ over $${\mathbb {F}}_{p^m}+u{\mathbb {F}}_{p^m}$$ $$(u^2=0)$$ . Also, we determine the RT (Rosenbloom–Tsfasman) distances, RT weight distributions, and Hamming distances of such constacyclic codes.