A Polynomial Algorithm for Codes Based on Directed Graphs

A complete description and proof of correctness are given for a new polynomial time algorithm for a class of codes based on directed graphs and involving construction well known in system theory. Our construction has already been considered in the literature in relation to other questions. The investigation of codes in this graph-based construction is inspired by analogy with classical cyclic codes that are defined in a similar way in polynomial rings. We show that all cyclic codes can be embedded in this construction. For each graph, the algorithm computes the largest number of errors which can be corrected by codes defined with this graph. In addition, it finds a generator of a code with this optimum value