On linear $q$-ary completely regular codes with $\rho=2$ and dual antipodal

TL;DR

This paper characterizes all linear q-ary completely regular codes with covering radius 2 whose duals are antipodal, extending known classifications for radius 1 and identifying all such codes, including two-weight antipodal codes.

Contribution

It provides a complete characterization of linear q-ary completely regular codes with radius 2 and antipodal duals, expanding the classification beyond radius 1.

Findings

All such codes with radius 2 are characterized.

The work includes a list of known codes with these properties.

It also characterizes two-weight linear antipodal codes.

Abstract

We characterize all linear $q$-ary completely regular codes with covering radius $\rho=2$ when the dual codes are antipodal. These completely regular codes are extensions of linear completely regular codes with covering radius 1, which are all classified. For $\rho=2$, we give a list of all such codes known to us. This also gives the characterization of two weight linear antipodal codes.