Explicit representation for a class of Type 2 constacyclic codes over the ring ${\mathbb{F}}_{2^m}[u]\mathrm{/}⟨u^{2\lambda }⟩$ with even length

Let ${\mathbb{F}}_{2^m}$ be a finite field of cardinality $2^m$, $\lambda$ and $k$ be integers satisfying $\lambda ,k\ge 2$ and denote $R={\mathbb{F}}_{2^m}[u]\mathrm{/}⟨u^{2\lambda }⟩$. Let $\delta,\alpha\in $\mathbb{F}$_{2^m}^{\times}$\mathrm{.}Foranyoddpositiveinteger$n$, we give an explicit representation and enumeration for all distinct $(\delta+\alpha u^2)$-constacycliccodesover$R$oflength$2^kn$, and provide a clear formula to count the number of all these codes. As a corollary, we conclude that every $(\delta +\alpha u^2)$-constacyclic code over $R$ of length $2^{k}n$ is an ideal generated by at most $2$ polynomials in the residue class ring $R[x]/\langle x^{2^kn}-(\delta+\alpha u^2)\rangle$.