In this paper, a class of additive codes which is referred to as \(\mathbb {Z}_{2}\mathbb {Z}_{2}[u,v]\)-additive codes is introduced. This is a generalization towards another direction of recently introduced \(\mathbb {Z}_{2}\mathbb {Z}_{4}\) codes (Doughterty et al., Appl. Algebra Eng. Commun. Comput. 27(2), 123–138, 7). A MacWilliams-type identity that relates the weight enumerator of a code with its dual is proved. Furthermore, the structure and possible weights for all one-weight and two-weight \(\mathbb {Z}_{2}\mathbb {Z}_{2}[u,v]\)-additive codes are described. Additionally, we also construct some one-weight and two-weight \(\mathbb {Z}_{2}\mathbb {Z}_{2}[u,v]\)-additive codes to illustrate our obtained results.

中文翻译：

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一重和二重ℤ2ℤ2[u，v] $ \ mathbb {Z} _ {2} \ mathbb {Z} _ {2} [u，v] $-加法码

本文介绍了一类加法码，称为\（\ mathbb {Z} _ {2} \ mathbb {Z} _ {2} [u，v] \）-加法码。这是朝着最近推出另一个方向的推广\（\ mathbb {Z} _ {2} \ mathbb {Z} _ {4} \）码（Doughterty等人，申请代数主机。COMMUN。COMPUT。27（ 2），123–138，7）。证明了将代码的权重枚举数与其对偶相关联的MacWilliams类型身份。此外，描述了所有一重和两重\（\ mathbb {Z} _ {2} \ mathbb {Z} _ {2} [u，v] \）加法码的结构和可能的权重。此外，我们还构造了一些一重和二重\（\ mathbb {Z} _ {2} \ mathbb {Z} _ {2} [u，v] \）-附加代码来说明我们获得的结果。