In this study we consider Euclidean and Hermitian self-dual codes over the direct product ring \(\mathbb {F}_{2} \times (\mathbb {F}_{2}+v\mathbb {F}_{2})\) where v² = v. We obtain some theoretical outcomes about self-dual codes via the generator matrices of free linear codes over \(\mathbb {F}_{2} \times (\mathbb {F}_{2}+v\mathbb {F}_{2})\). Also, we obtain upper bounds on the minimum distance of linear codes for both the Lee distance and the Gray distance. Moreover, we find some free Euclidean and free Hermitian self-dual codes over \(\mathbb {F}_{2} \times (\mathbb {F}_{2}+v\mathbb {F}_{2})\) via some useful construction methods.

在这项研究中，我们考虑直接乘积环\（\ mathbb {F} _ {2} \ times（\ mathbb {F} _ {2} + v \ mathbb {F} _ {2 }）\）其中v ² = v。我们通过\（\ mathbb {F} _ {2} \ times（\ mathbb {F} _ {2} + v \ mathbb {F} _ {2}）\）。同样，对于李距离和格雷距离，我们都获得了线性编码最小距离的上限。而且，我们在\（\ mathbb {F} _ {2} \ times（\ mathbb {F} _ {2} + v \ mathbb {F} _ {{2}）上找到一些免费的欧几里得和厄米自对偶代码\）通过一些有用的构造方法。