Characterizing completely regular codes from an algebraic viewpoint

Understand

We first summarize the basic structure of the outer distribution module of a completely regular code.

• Then, employing a simple lemma concerning eigenvectors in association schemes, we propose to study the tightest case, where the indices of the eigenspace that appear in the outer distribution module are equally spaced.
• In addition to the arithmetic codes of the companion paper, this highly structured class includes other beautiful examples and we propose the classification of $Q$-polynomial completely regular codes in the Hamming graphs.
• A key result is Theorem 3.10 which finds that the $Q$-polynomial condition is equivalent to the presence of a certain Leonard pair.