Association schemes related to universally optimal configurations, Kerdock codes and extremal Euclidean line-sets

TL;DR

This paper explores association schemes linked to universally optimal configurations, Kerdock codes, and extremal Euclidean line-sets, revealing new connections and dualities in algebraic combinatorics and coding theory.

Contribution

It generalizes a known association scheme in R^{14} using Kerdock codes and mutually unbiased bases, and explains formal duality through dual abelian schemes related to Kerdock and Preparata codes.

Findings

Established a connection between association schemes and Kerdock codes.

Constructed dual abelian schemes related to quaternary codes.

Linked association schemes to extremal line-sets and Barnes-Wall lattices.

Abstract

H. Cohn et. al. proposed an association scheme of 64 points in R^{14} which is conjectured to be a universally optimal code. We show that this scheme has a generalization in terms of Kerdock codes, as well as in terms of maximal real mutually unbiased bases. These schemes also related to extremal line-sets in Euclidean spaces and Barnes-Wall lattices. D. de Caen and E. R. van Dam constructed two infinite series of formally dual 3-class association schemes. We explain this formal duality by constructing two dual abelian schemes related to quaternary linear Kerdock and Preparata codes.