Minimum distance of Coxeter codes

Determine the minimum Hamming distance of the order-r Coxeter code of a finite Coxeter system (W,S) of rank m, and prove that it equals \(\min_{J\subseteq S,\,|J|=m-r}|J|\).

Background

The paper defines the order-r Coxeter code associated with a finite Coxeter system (W,S) of rank m as the binary linear span of the indicators of all standard cosets of rank m-r. The paper establishes structural properties of these codes, including nestedness, duality, and a dimension formula in terms of the W-Eulerian numbers, but does not determine their minimum distance in general.

The stated conjecture proposes an explicit formula for the distance. It is known for ReedMuller codes and for Coxeter codes arising from dihedral groups (I_2(n)) for all n\geq2, and the authors report computer verification for all nontrivial Coxeter codes of length at most 120.

References

The distance of ${r}$ is given by

\mathrm{dist}({r})=\min_{J\subseteq S, \abs{J}=m-r} \abs{J}.

This conjecture is known to be true for RM codes and the family of Coxeter codes given by the dihedral groups, $I_2(n)$, for all $n\geq 2$.

— Coxeter codes: Extending the Reed-Muller family  (2502.14746 - Coble et al., 20 Feb 2025) in Conjecture 1, Section 3.2 (Rate of the codes)