Abstract
In this work, we first generalize the $$\sigma $$-LCD codes over finite fields to $$\sigma $$-LCD codes over finite chain rings. Under suitable conditions, linear codes over finite chain rings that are $$\sigma $$-LCD codes are characterized. Then we provide a necessary and sufficient condition for free constacyclic codes over finite chain rings to be $$\sigma $$-LCD. We also get some new binary LCD codes of different lengths which come from Gray images of constacyclic $$\sigma $$-LCD codes over $$\mathbb {F}_{2}+\gamma \mathbb {F}_{2}+\gamma ^2\mathbb {F}_{2}$$. Finally, for special finite chain rings $$\mathbb {F}_{q}+\gamma \mathbb {F}_{q}$$, we define a new Gray map $$\Phi $$ from $$(\mathbb {F}_{q}+\gamma \mathbb {F}_{q})^n$$ to $$\mathbb {F}_{q}^{2n}$$, and by using $$\sigma $$-LCD codes over finite chain rings $$\mathbb {F}_{q}+\gamma \mathbb {F}_{q}$$, we construct new entanglement-assisted quantum error-correcting (abbreviated to EAQEC) codes with maximal entanglement and parts of them are MDS EAQEC codes.
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