In this paper, we introduce \({\mathbb {F}}_{q^{2}}\)-double cyclic codes of length \(n=r+s,\) where \({\mathbb {F}}_{q^{2}}\) is the Galois field of \(q^{2}\) elements, q is a power of a prime integer p and r, s are positive integers. We determine the generator polynomials for any \({{\mathbb {F}}_{q^{2}} }\)-double cyclic code. For any \({\mathbb {F}}_{q^{2}}\)-double cyclic code \({\mathcal {C}},\) we will define the Euclidean dual code \({\mathcal {C}}^{\perp }\) based on the Euclidean inner product and the Hermitian dual code \({\mathcal {C}} ^{\perp _{H}}\) based on the Hermitian inner product. We will construct a relationship between \({\mathcal {C}}^{\perp }\) and \({\mathcal {C}}^{\perp _{H}}\) and then find the generator polynomials for the Hermitian dual code \({\mathcal {C}}^{\perp _{H}}.\) As an application of our work, we will present examples of optimal parameter linear codes over the finite field \({\mathbb {F}} _{4}\) and also examples of optimal quantum codes that were derived from \({\mathbb {F}}_{4}\)-double cyclic codes using the Hermitian inner product.

在本文中，我们介绍\({\mathbb {F}}_{q^{2}}\) -长度为\(n=r+s,\) 的双循环码，其中\({\mathbb {F} }_{q^{2}}\)是\(q^{2}\)元素的伽罗瓦域，q是素数p和r的幂，  s是正整数。我们确定任何\({{\mathbb {F}}_{q^{2}} }\) -double 循环码的生成多项式。对于任何\({\mathbb {F}}_{q^{2}}\) -双循环码\({\mathcal {C}},\)，我们将定义欧几里得对偶码\({\mathcal { C}}^{\perp }\)基于欧几里得内积和 Hermitian 对偶码\({\mathcal {C}} ^{\perp _{H}}\)基于 Hermitian 内积。我们将构建\({\mathcal {C}}^{\perp }\)和\({\mathcal {C}}^{\perp _{H}}\) 之间的关系，然后找到生成多项式Hermitian 对偶码\({\mathcal {C}}^{\perp _{H}}.\)作为我们工作的一个应用，我们将展示有限域\({\mathbb {F}} _{4}\)以及从\({\mathbb {F}}_{4}\) - 使用 Hermitian 内积的双循环码派生的最佳量子码示例。