A class of repeated-root constacyclic codes over 𝔽_p^m [u]/〈u^e〉 of Type 2

Let  be a finite field of cardinality  where p is an odd prime, n be a positive integer satisfying , and denote  where  be an even integer. Let . Then the class of constacyclic codes over R is a significant subclass of constacyclic codes over R of Type 2. For any integer , an explicit representation and a complete description for all distinct constacyclic codes over R of length np^k and their dual codes are given. Moreover, formulas for the number of codewords in each code and the number of all such codes are provided respectively. In particular, all distinct constacyclic codes over  of length p^k and their dual codes are presented precisely.