Construction of quantum codes from multivariate polynomial rings

In this paper, we use multivariate polynomial rings to construct quantum error-correcting codes (QECCs) via Hermitian construction. We establish a relation between linear codes and ideals of multivariate polynomial rings. We give a necessary and suffcient condition for a multivariate polynomial to generate a Hermitian dual-containing code. By comparing with the literatures in recent years, we construct 31 new QECCs over \(\mathbb {F}_q\) ${\mathbb{F}}_q$ , where \(q=3,4,5,7\) $q=3,4,5,7$ . Some of them reach quantum singleton bound and some of them exceed quantum GV bound.