A study of QECCs and EAQECCs construction from cyclic codes over the ring \({\mathbb {F}}_q+v_1{\mathbb {F}}_q+v_2{\mathbb {F}}_q+\cdots +v_s{\mathbb {F}}_q\)

In this paper, we present a construction of quantum error-correcting codes (QECCs) codes and entanglement-assisted quantum error-correcting (EAQECCs) using Euclidean hulls and sums of cyclic codes of length n over a family of ring \(R_s = {\mathbb {F}}_q+v_1{\mathbb {F}}_q+v_2{\mathbb {F}}_q+\cdots +v_s{\mathbb {F}}_q\) $R_s={\mathbb{F}}_q+v_1{\mathbb{F}}_q+v_2{\mathbb{F}}_q+\cdots +v_s{\mathbb{F}}_q$ , where q is an odd prime power and \(v_i ^2=v_i\) $v_i^2=v_i$ , \(v_iv_j=v_jv_i=0\) $v_i{v}_j=v_j{v}_i=0$ , for \(i,j= 1,2,3,\cdots ,s\) $i,j=1,2,3,\cdots ,s$ and \(i \ne j\) $i\ne j$ . The study delves into various aspects of this construction. We explore the generator polynomials, the dimension of both Euclidean hulls and the sums of cyclic codes over the ring \(R_s\) $R_s$ . Further, we determine several new QECCs and EAQECCs. This paper claims that our obtained codes have improved parameters (e.g. higher minimum distance or greater dimension) than the existing quantum codes. Moreover, we present some detailed examples that effectively illustrate our findings.