Codes and designs in multivariate $Q$-polynomial association schemes

TL;DR

This paper extends classical bounds on codes and designs within multivariate Q-polynomial association schemes, introducing new bounds, characterizations, and applications to various combinatorial structures.

Contribution

It generalizes Delsarte's bounds to weakly metric schemes, characterizes optimal codes and designs via Wilson polynomials, and explores duality and applications in combinatorics.

Findings

Upper bounds on code sizes with given degree or pairwise distances

Characterization of codes meeting bounds via Wilson polynomial annihilators

Existence of tight designs implies Lloyd-like polynomial conditions

Abstract

We generalize the fundamental bounds of Delsarte thesis (1973) on codes of given degree and designs of given strength in the new setting of Bannai et al. (2025). We assume the scheme is weakly metric in the sense of (Sol\'e, 1989). We give upper bounds on the size of codes of given degree, and also on the size of codes with a given number of pairwise distances. Codes meeting these bounds are characterized by the identification of suitable annihilators with the degree (resp. distance) Wilson polynomial. We give two analogues of the Rao bound on the size of designs with given strength. Designs meeting that bound we call degree tight designs or distance tight design depending on the bound met. In both cases, the existence of a tight design implies a Lloyd-like condition on a suitable analogue of the Wilson polynomial. Applications to the Lee distance, mixed level orthogonal arrays, ordered orthogonal arrays, and more are given. The formal duality between codes and designs, connecting perfect codes and tight designs, is made concrete in self-dual translation schemes.