Reversible Codes in <i> Mat _n×s </i> (F <i>q</i> ) Under NRT-Metric

Reversible codes play an essential role in coding theory because of their structural symmetry and broad applications in error control, cryptography, and DNA computing. This study investigates reversible codes over the matrix space <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Mat_n×s</i>(F<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>q</i></sub>) equipped with the Niederreiter–Rosenbloom–Tsfasman (NRT) metric. We introduce the concepts of reversible and self-reversible codes in the NRT-metric and establish several structural properties of these codes. Explicit conditions for the reversibility of linear codes based on generator matrices were derived for both the even and odd dimensions. We further prove that the duals of NRT-reversible codes are also NRT-reversible and present characterizations linking the number of self-reversible codewords with the block structure of generator matrices. The necessary and sufficient condition for a self-dual code to be reversible is proven for both the even and odd dimensions. Several examples of codes over finite fields illustrate and demonstrate the construction of NRT-reversible codes.