Families of nested completely regular codes and distance-regular graphs

In this paper infinite families of linear binary nested completely regular codes are constructed. They have covering radius ρ equal to 3 or 4, and are 1/2-th parts, for i ∈ {1, . . . , u} of binary (respectively, extended binary) Hamming codes of length n = 2 − 1 (respectively, 2), where m = 2u. In the usual way, i.e., as coset graphs, infinite families of embedded distance-regular coset graphs of diameter D equal to 3 or 4 are constructed. In some cases, the constructed codes are also completely transitive codes and the corresponding coset graphs are distance-transitive. ∗This work has been partially supported by the Spanish MICINN grant TIN201340524-P; the Catalan grant 2009SGR1224 and also by the Russian fund of fundamental researches 12-01-00905. J. Rifà and J. Borges are with the Department of Information and Communications Engineering, Universitat Autònoma de Barcelona. 2 V. Zinoviev is with the A. A. Harkevich Institute for Problems of Information Transmission, Russian Academy of Sciences.