In this paper, we first generalize the polycyclic codes over finite fields to polycyclic codes over the mixed alphabet \(\mathbb {Z}_2\mathbb {Z}_4\), and we show that these codes can be identified as \(\mathbb {Z}_4[x]\)-submodules of \(\mathcal {R}_{\alpha ,\beta }\) with \(\mathcal {R}_{\alpha ,\beta }=\mathbb {Z}_2[x]/\langle t_1(x)\rangle \times \mathbb {Z}_4[x]/\langle t_2(x)\rangle \), where \(t_1(x)\) and \(t_2(x)\) are monic polynomials over \(\mathbb {Z}_2\) and \(\mathbb {Z}_4\), respectively. Then we provide the generator polynomials and minimal generating sets for this family of codes based on the strong Gröbner basis. In particular, under the proper defined inner product, we study the dual of \(\mathbb {Z}_2\mathbb {Z}_4\)-additive polycyclic codes. Finally, we focus on the characterization of the \(\mathbb {Z}_2\mathbb {Z}_4\)-MDSS and MDSR codes, and as examples, we also present some (almost) optimal binary codes derived from the \(\mathbb {Z}_2\mathbb {Z}_4\)-additive polycyclic codes.

在本文中，我们首先将有限域上的多环码推广到混合字母表\(\mathbb {Z}_2\mathbb {Z}_4\)上的多环码，我们证明这些码可以识别为\(\ mathbb {Z}_4[x]\) - \(\mathcal {R}_{\alpha ,\beta }\) 的子模块与\(\mathcal {R}_{\alpha ,\beta }=\mathbb { Z}_2[x]/\langle t_1(x)\rangle \times \mathbb {Z}_4[x]/\langle t_2(x)\rangle \)，其中\(t_1(x)\)和\( t_2(x)\)是\(\mathbb {Z}_2\)和\(\mathbb {Z}_4\)上的单项多项式， 分别。然后我们基于强 Gröbner 基础为这组代码提供生成多项式和最小生成集。特别是，在适当定义的内积下，我们研究了\(\mathbb {Z}_2\mathbb {Z}_4\) -加性多环码的对偶。最后，我们专注于\(\mathbb {Z}_2\mathbb {Z}_4\) -MDSS 和 MDSR 代码的表征，并且作为示例，我们还提供了一些（几乎）从\( \mathbb {Z}_2\mathbb {Z}_4\) -加性多环码。