The Griesmer codes of Belov type and optimal quaternary codes via multi-variable functions

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.