Quantum synchronizable codes from the ring \(\mathbb {F}_{q} +\) \(\varvec{v}\) \(\mathbb {F}_{q}\)

Let \(R_{v,q}=\mathbb {F}_{q}+v\mathbb {F}_{q}\) $R_{v,q}={\mathbb{F}}_q+v{\mathbb{F}}_q$ with \(v^2=v\) $v^2=v$ . In this paper, we provide three methods of constructing quantum synchronizable codes (QSCs) by using the Euclidean sums of \((1-2v)\) $(1-2v)$ -constacyclic codes over \(R_{v,q}\) $R_{v,q}$ . Concrete examples are presented to enrich the variety of available QSCs. In addition, two of methods in our research are easier than available methods for constructing QSCs and producing more QSCs.