Completely regular codes – a viewpoint and some problems

Since their introduction in 1973, completely regular codes have been of interest to coding theorists and graph theorists alike. These highly regular substructures were defined as a generalization of perfect and uniformly packed error-correcting codes but also include many codes having very small minimum distance which are fundamental to the study of distance-regular graphs. While interest in these codes among coding theorists seems to be on the decline, there is reason to believe that the importance of completely regular codes to the theory of distance-regular graphs has yet to be fully realized. This paper is an attempt to tell this story and explain these trends. Some open problems are discussed and a bibliography of recent literature is included.