We study the Griesmer codes of specific Belov type and construct families of distance-optimal linear codes over \({\mathbb {Z}_4}\) by using multi-variable functions. We first show that the pre-images of specific Griesmer codes of Belov type under a Gray map \(\phi \) from \({\mathbb {Z}_4}\) to \(\mathbb {Z}_2^2\) are non-linear except one case. Therefore, we are interested in finding subcodes of Griesmer codes of specific Belov type with maximum possible dimension whose pre-images under \(\phi \) are still linear over \({\mathbb {Z}_4}\) such that they also have good properties such as optimality and two-weight. To this end, we introduce a new approach for constructing linear codes over \({\mathbb {Z}_4}\) using multi-variable functions over \(\mathbb {Z}\). This approach has an advantage in explicitly computing the Lee weight enumerator of a linear code over \({\mathbb {Z}_4}\). Furthermore, we obtain several other families of distance-optimal two-weight linear codes over \({\mathbb {Z}_4}\) by using a variety of multi-variable functions. We point out that some of our families of distance-optimal codes over \({\mathbb {Z}_4}\) have linear binary Gray images which are also distance-optimal.

我们研究特定 Belov 类型的 Griesmer 码，并通过使用多变量函数在\({\mathbb {Z}_4}\)上构造距离最优线性码族。我们首先展示了 Belov 类型的特定 Griesmer 码在灰度映射\(\phi \)下从\({\mathbb {Z}_4}\)到\(\mathbb {Z}_2^2\)的原像）除一种情况外都是非线性的。因此，我们有兴趣找到具有最大可能维度的特定 Belov 类型的 Griesmer 码的子码，其在\(\phi \)下的原像在\({\mathbb {Z}_4}\)上仍然是线性的，这样它们也具有良好的性质，例如最优性和二重性。为此，我们引入了一种使用\(\mathbb {Z}\)上的多变量函数在\({\mathbb {Z}_4}\)上构造线性代码的新方法。与\({\mathbb {Z}_4}\)相比，此方法在显式计算线性代码的 Lee 权重枚举器方面具有优势。此外，我们通过使用各种多变量函数，在\({\mathbb {Z}_4}\)上获得了其他几个系列的距离最优二权重线性码。我们指出，我们在\({\mathbb {Z}_4}\)上的一些距离最优代码系列具有线性二值灰度图像，它们也是距离最优的。