Existence of Lehmer codes for the remaining finite irreducible Coxeter systems

Determine whether Lehmer codes exist for finite irreducible Coxeter systems not covered by the constructions given for types A_n, B_n, D_n, H_3, and I_2(m).

Background

The paper defines a Lehmer code for a finite Coxeter system (W,S) as a bijection from W to the product of chains determined by the exponents of (W,S), whose inverse is a poset morphism from the componentwise order on that product to the Bruhat order on W. Such a code realizes every lower Bruhat interval as a multicomplex and yields a shellable, vertex-decomposable Lehmer complex whose h-polynomial is the interval's rank-generating function.

Explicit Lehmer codes are constructed in the paper for several families, including types A_n, B_n, D_n, H_3, and I_2(m). The existence question for finite irreducible Coxeter systems outside these established cases remains unresolved, and its resolution would extend the paper's combinatorial construction and its consequences for Bruhat intervals to additional finite Coxeter groups.