\({\mathbb {Z}}_2{\mathbb {Z}}_{4}\)-additive codes have been defined as a subgroup of \({\mathbb {Z}}_2^{r}\times {\mathbb {Z}}_4^{s}\) in [6] where \({\mathbb {Z}}_2\), \({\mathbb {Z}}_{4}\) are the rings of integers modulo 2 and 4 respectively and r and s are positive integers. In this study, we define a family of codes over the set \({\mathbb {Z}}_2[{\bar{\xi }}]^{r}\times {\mathbb {Z}}_4[\xi ]^{s}\) where \(\xi \) is a root of a monic basic primitive polynomial in \({\mathbb {Z}}_{4}[x]\). We give the standard form of the generator and parity-check matrices of codes over \({\mathbb {Z}}_2[{\bar{\xi }}]^{r}\times {\mathbb {Z}}_4[\xi ]^{s}\) and also we introduce skew cyclic codes and their spanning sets. Moreover, we study the Gray images of codes over both \({{\mathbb {Z}}}_4[\xi ]\) and \({{\mathbb {Z}}_{2}[{\bar{\xi }}]^r\times {{\mathbb {Z}}_{4}[\xi ]^s}}\) with respect to homogeneous weight and give the necessary and sufficient condition for their Gray images to be a linear code. We further present some examples of optimal codes which are actually Gray images of skew cyclic codes.

\({\mathbb {Z}}_2{\mathbb {Z}}_{4}\) -可加码被定义为\({\mathbb {Z}}_2^{r}\times { \mathbb {Z}}_4^{s}\)在 [6] 中\({\mathbb {Z}}_2\) , \({\mathbb {Z}}_{4}\)是整数分别模 2 和 4，r和s是正整数。在本研究中，我们在集合\({\mathbb {Z}}_2[{\bar{\xi }}]^{r}\times {\mathbb {Z}}_4[\xi ]^{s}\)其中\(\xi \)是\({\mathbb {Z}}_{4}[x]\)中的单调基本本原多项式的根。我们给出了代码的生成器和奇偶校验矩阵的标准形式\({\mathbb {Z}}_2[{\bar{\xi }}]^{r}\times {\mathbb {Z}}_4[\xi ]^{s}\)并且我们还引入了偏斜循环代码及其生成集。此外，我们研究了\({{\mathbb {Z}}}_4[\xi ]\)和\({{\mathbb {Z}}_{2}[{\bar{\ xi }}]^r\times {{\mathbb {Z}}_{4}[\xi ]^s}}\)关于齐次权重，并给出其灰度图像为线性的充要条件代码。我们进一步介绍了一些最佳代码的例子，它们实际上是倾斜循环代码的格雷图像。