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

Abstract

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.