MacWilliams Duality in \(LRTJ\)-Spaces

Abstract

In [4], the author introduced a new metric on the space \(\text{Mat}_{m \times s}(\mathbb{Z}_q)\), which is the module space of all \(m \times s\) matrices with entries from the finite ring \(\mathbb{Z}_q\) (\(q \geq 2\)), generalizing the classical Lee metric [5] and the array RT-metric [8], and named this metric as GLRTP-metric, which is further renamed as LRTJ-metric (Lee-Rosenbloom-Tsfasman-Jain Metric) in [1]. In this paper, we introduce a complete weight enumerator for codes over \(\text{Mat}_{m \times s}(\mathbb{Z}_q)\) endowed with the LRTJ-metric and obtain a MacWilliams-type identity with respect to this new metric for the complete weight enumerator.