Arithmetic completely regular codes
Jacobus H. Koolen, Woo Sun Lee, William J. Martin, Hajime Tanaka
Abstract
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). Connections to the theory of distance-regular graphs are
explored and several open questions are posed.
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