# Characterizing completely regular codes from an algebraic viewpoint

## TL;DR

This paper explores the algebraic structure of completely regular codes in association schemes, especially Q-polynomial codes, revealing their connection to Leonard pairs and classifying these codes within Hamming graphs.

## Contribution

It introduces a new algebraic characterization of Q-polynomial completely regular codes via Leonard pairs and classifies these codes in Hamming graphs.

## Key findings

- Q-polynomial codes are characterized by Leonard pairs.
- The classification of Q-polynomial codes in Hamming graphs is achieved.
- New connections between Leonard pairs and association schemes are established.

## Abstract

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. This connection has impact in two directions. First, the Leonard pairs are classified and we gain quite a bit of information about the algebraic structure of any code in our class. But also this gives a new setting for the study of Leonard pairs, one closely related to the classical one where a Leonard pair arises from each thin/dual-thin irreducible module of a Terwilliger algebra of some $P$- and $Q$-polynomial association scheme, yet not previously studied. It is particularly interesting that the Leonard pair associated to some code $C$ may belong to one family in the Askey scheme while the distance-regular graph in which the code is found may belong to another.