Arithmetic completely regular codes

Understand

In this paper, we explore completely regular codes in the Hamming graphs and related graphs.

• Experimental evidence suggests that many completely regular codes have the property that the eigenvalues of the code are in arithmetic progression.
• In order to better understand these "arithmetic completely regular codes", we focus on cartesian products of completely regular codes and products of their corresponding coset graphs in the additive case.
• Employing earlier results, we are then able to prove a theorem which nearly classifies these codes in the case where the graph admits a completely regular partition into such codes (e.g, the cosets of some additive completely regular code).