We construct a family of cyclic completely regular codes (CRCs) of length \(n=2^m-1\) and compute their intersection arrays. These codes, denoted \(C_{1,5}\) , are generated by the product \(m_1(x)m_5(x)\) , where \(m_i(x)\) is the minimal polynomial of \(\alpha ^i\) , and \(\alpha \) is a primitive element of the finite field \(\mathbb {F}_{2^m}\) . We consider two main cases: odd m and \(m \equiv 2 \pmod {4}\) . For odd m, these codes are known to be completely regular with covering radius \(\rho =3\) and minimum distance \(d=5\) . We prove that, for any m, the codes \(C_{1,3}\) and \(C_{1,5}\) are non-equivalent despite sharing the same parameters and intersection array. For \(m \equiv 2 \pmod {4}\) , we demonstrate that \(C_{1,5}\) forms a new family of completely regular codes with covering radius \(\rho =3\) , minimum distance \(d=3\) , and intersection array \(\operatorname {IA}=[n, n-3, \frac{3n+7}{4}; 1, 4, \frac{n-3}{4}]\) . Moreover, the corresponding extended cyclic codes \(C_{1,5}^*\) are completely regular \([n+1, n-2m, 4; 4]\) -codes with intersection array \(\operatorname {IA}=[n+1, n, n-3, \frac{3n+7}{4}; 1, 4, \frac{n-3}{4}, n+1]\) . We also describe the parameters and some properties of the coset graphs associated to these completely regular codes which form distance-regular graphs.
