# American Institute of Mathematical Sciences

May  2015, 9(2): 233-246. doi: 10.3934/amc.2015.9.233

## Families of nested completely regular codes and distance-regular graphs

 1 Department of Information and Communications Engineering, Universitat Autònoma de Barcelona, 08193 Bellaterra, Cerdanyola del Vallès, Spain 2 Department of Information and Communications Engineering, Universitat Autònoma de Barcelona, 08193-Bellaterra, Cerdanyola del Vallès 3 A. A. Kharkevich Institute for Problems of Information Transmission, Russian Academy of Sciences, GSP-4, Moscow, 127994, Russian Federation

Received  June 2014 Published  May 2015

In this paper infinite families of linear binary nested completely regular codes are constructed. They have covering radius $\rho$ equal to $3$ or $4,$ and are $1/2^i$th parts, for $i\in\{1,\ldots,u\}$ of binary (respectively, extended binary) Hamming codes of length $n=2^m-1$ (respectively, $2^m$), 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. This gives antipodal covers of some distance-regular and distance-transitive graphs. In some cases, the constructed codes are also completely transitive and the corresponding coset graphs are distance-transitive.
Citation: Joaquim Borges, Josep Rifà, Victor A. Zinoviev. Families of nested completely regular codes and distance-regular graphs. Advances in Mathematics of Communications, 2015, 9 (2) : 233-246. doi: 10.3934/amc.2015.9.233
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