Let \(\mathcal {R}=\mathbb {F}_{q^2}+\mu \mathbb {F}_{q^2}+\nu \mathbb {F}_{q^2}\) , where q is a prime power, \(\mu ^2=\mu \) , \(\nu ^2=\nu \) and \(\mu \nu =\nu \mu =0\) . In this paper, we study the structural properties of skew \(\lambda \) -constacyclic codes over \(\mathbb {F}_{q^2}\mathcal {R}\) , where \(\lambda \) is a unit of \(\mathcal {R}\) . Define a Gray map \(\varPhi \) from \(\mathbb {F}_{q^2}^{\alpha }\times \mathcal {R}^{\beta }\) to \(\mathbb {F}_{q^2}^{\alpha +3\beta }\) preserving the Hermitian orthogonality, where \(\alpha \) and \(\beta \) are positive integers. Furthermore, we give a necessary and sufficient condition for skew \(\lambda \) -constacyclic codes over \(\mathbb {F}_{q^2}\mathcal {R}\) to be linear complementary dual (LCD). Then we obtain some LCD codes as the \(\varPhi \) -images of Hermitian LCD skew \(\lambda \) -constacyclic codes. We also enumerate the Hermitian LCD skew \(\lambda \) -constacyclic codes over \(\mathcal {R}\) . Finally, based on the study of Hermitian LCD skew \(\lambda \) -constacyclic codes over \(\mathbb {F}_{q^2}\mathcal {R}\) , we employ the Gray map to construct entanglement-assisted quantum error-correcting codes (EAQECCs) with the maximal entanglement.
