We obtain a classification of the completely regular codes with covering radius 1 and the second eigenvalue in the Hamming graphs H(3,q) up to q and intersection array. Due to works of Meyerowitz, Mogilnykh and Valyuzenich, our result completes the classifications of completely regular codes with covering radius 1 and the second eigenvalue in the Hamming graphs H(n,q) for any n and completely regular codes with covering radius 1 in H(3,q).
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Mogilnykh et al. (2024) studied this question.