Matrix graphs and MRD codes over finite principal ideal rings

摘要

Let R be a finite principal ideal ring and m, n, d positive integers. In this paper, we study the matrix graph over R which is the graph whose vertices are m x n matrices over Rand two matrices A and Bare adjacent if and only if 0 rank(A-B) d. We show that this graph is a connected vertex transitive graph. The distance, diameter, independence number, clique number and chromatic number of this graph are also determined. This graph can be applied to study MRD codes over R. We obtain that a maximal independent set of the matrix graph is a maximum rank distance (MRD) code and vice versa. Moreover, we show the existence of linear MRD codes over R. (C) 2020 Elsevier Inc. All rights reserved.