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.
Highlights
In this paper, we present the theory of completely regular codes in the Hamming graph enjoying the property that the eigenvalues of the code are in arithmetic progression
The class of arithmetic codes we introduce in this paper is perhaps the most important subclass of the Leonard completely regular codes in the Hamming graphs and something similar is likely true for the other classical families, but this investigation is left as an open problem
First noting (Proposition 3.1) that a completely regular product must arise from completely regular constituents, we determine in Proposition 3.4 exactly when the product of two completely regular codes in two Hamming graphs is completely regular
Summary
We present the theory of completely regular codes in the Hamming graph enjoying the property that the eigenvalues of the code are in arithmetic progression. First noting (Proposition 3.1) that a completely regular product must arise from completely regular constituents, we determine in Proposition 3.4 exactly when the product of two completely regular codes in two Hamming graphs is completely regular At this point, the role of the arithmetic property becomes clear and we understand that Lemma 3.3 gives a generic form for the quotient matrix of such a code. When C is a linear completely regular code with the arithmetic property and C has minimum distance at least three and covering radius at most two, we show that C is closely related to some Hamming code These results are summarized in Theorem 3.16, which gives a full classification of possible codes and quotients in the linear case (always assuming the arithmetic and completely regular properties) and Corollary 3.17 which characterizes Hamming quotients of Hamming graphs
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