### On cyclic codes in incidence rings

Ricardo Alfaro, Andrei V. Kelarev
2006 Studia scientiarum mathematicarum Hungarica (Print)
Cyclic codes are defined as ideals in polynomial quotient rings. We are using a matrix ring construction in a similar way to define classes of codes. It is shown that all cyclic and all linear codes can be embedded as ideals in this construction. A formula for the largest Hamming weight of one-sided ideals in incidence rings is given. It is shown that every incidence ring defined by a directed graph always possesses a principal one-sided ideal that achieves the optimum Hamming weight. It is
more » ... known that various classical error-correcting codes are ideals in certain algebras. For example, all cyclic codes are principal ideals in group algebras of cyclic groups. Several other classes of codes have also been shown to be ideals in group algebras, and this additional algebraic structure has been used to develop faster encoding and decoding algorithms for these codes (see, for example, , , ). The investigation of weights of ideals in other ring constructions was begun in  and is related to the general Problem 10.1 in  . The aim of this paper is to use the construction of incidence rings in a similar way. We demonstrate that all cyclic codes and all linear codes can be embedded in incidence rings of graphs as ideals. A new formula is given for the maximum Hamming weights of one-sided ideals in incidence rings. It is shown that, for this construction, there always exist principal ideals achieving the optimal value. Throughout the word 'graph' means a directed graph without multiple edges but possibly with loops. As it is customary in coding theory, let F be a finite field regarded as an encoding alphabet, 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 following definition of an incidence ring (see, for example, , §3.15). The 0 Mathematics Subject Classification (1991): Primary: 94B60, 94B65; Secondary: 16S50.