New Bounds of Linear Matrix Codes for the Rosenbloom-Tsfasman Metric and Optimal Constructions

The Rosenbloom-Tsfasman metric (RT-metric for short) is a generalization of the Hamming metric. Matrix codes in the frame of the RT-metric have been used in information transmission over parallel channels. In this paper, we develop some new upper bounds on the minimum RT-distance of an $[h \times n, k, d_{\mathrm {RT}}]$ linear matrix code, which generalize the Singleton-type bound derived by Rosenbloom and Tsfasman. It should be emphasized that the upper bounds build a connection between the RT-metric and the Hamming metric. Constructions of linear matrix codes are presented and their parameters for the RT-metric are investigated. It is shown that every linear matrix code can be expressed by using the trace function, which is a generalization of the well-known defining-set construction of linear codes. Moreover, we obtain several classes of optimal linear matrix codes in this paper.