Directed Graphs and Minimum Distances of Error–correcting Codes in Matrix Rings

The main theorems of this paper give sharp upper bounds for the minimum distances of one-sided ideals in structural matrix rings defined by directed graphs. It is very well known that additional algebraic structure can give advantages for coding applications (see, for example, [8]). Serious attention in the literature has been devoted to considering properties of ideals in various ring constructions essential from the point of view of coding theory (see the survey [7] and books [4], [9], [10]). The investigation of code properties of ideals in structural matrix rings of directed graphs was begun in [6], where two-sided ideals are considered. The aim of this paper is to strengthen the results of [6] and obtain sharp upper bounds for the minimum distances of one–sided ideals in structural matrix rings defined by directed graphs. Let F be a finite field. Throughout, the word graph means a directed graph without multiple edges but possibly with loops, and D = (V, E) stands for a graph with the set V = {1, 2, . . . , n} of vertices and the set E of edges. Edges of D correspond to the standard elementary matrices of the algebra Mn(F ) of all (n×n)– matrices over F . Namely, for (i, j) ∈ E ⊆ V × V , let e(i,j) = ei,j = eij be the standard elementary matrix. Note that ei,jek,l = { 0 if j ̸= k, ei,l if j = k. Denote by MD(F ) = ⊕