The extended code for a class of generalized Roth-Lempel codes and their properties

As we all know, many interesting and important codes are obtained by modifying or combining existing codes. In this paper, we focus on generalized Roth-Lempel (in short, GRL) codes and define a class of extended codes, i.e., the extended generalized Roth-Lempel (in short, EGRL) code. And then for a special class of EGRL codes, we give a parity-check matrix and establish a necessary and sufficient condition for the EGRL code or its dual code to be MDS or AMDS, respectively. Finally, we construct a class of NMDS EGRL codes which is the generalization of the constructions given by Han et al. in 2023, and then completely determine its weight distribution.