Minimum Distances of Error-correcting Codes in Incidence Rings

It is very well known that additional algebraic structure can give advantages for coding applications. For example, all cyclic error-correcting codes are principal ideals in the group algebras of cyclic groups (see the survey [4] and the books [3, 5, 6, 7]). 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. The aim of this paper is to obtain a formula for the largest minimum distance of ideals in incidence rings defined by directed graphs. Let R be a ring with identity element 1, and let D = (V ,E) be any graph with the set V = {1, . . . ,n} of vertices and a set E ⊆ V ×V of edges. We use the standard definition of an incidence ring (see, e.g., [3, Section 3.15]). The incidence ring I(D,R) is the free left R-module with basis consisting of all edges in E, where multiplication is defined by the distributive law and the rule