Type 2 constacyclic codes over 𝔽_2^m [u]/⟨u³⟩ of oddly even length

Let  be a finite field of cardinality  be an odd positive integer, and denote . Let . Then  constacyclic codes over  are called constacyclic codes over  of Type 2. In this paper, an explicit representation and a complete description for all distinct  constacyclic codes over  of length  and their dual codes are given. Moreover, explicit formulas for the number of codewords in each code and the number of all such codes are provided respectively. In particular, all distinct self-dual constacyclic codes over  of length  are presented precisely. In addition, a complement to a result in Cao et al. (2017) is given.