Coxeter code[1]

Description

A member of a family of codes that generalizes RM codes in a group-theoretic sense.

A binary RM\((r,m)\) code is spanned by indicators of all subcubes of dimension \(m-r\) in the \(m\)-dimensional cube (this is a redundant generating set [2]), i.e., by the cosets of rank-\((m-r)\) subgroups of \(\mathbb{Z}_2^m\). For a finite Coxeter group \(W\) with \(m\) generators, a binary linear code \(C_W(r)\) of order \(r\) with \(-1 \leq r \leq m\) is similarly spanned by indicators of all the cosets of rank-\((m-r)\) parabolic subgroups of \(W\).

The dimension \(dim(C_W(r))=\sum_{i=0}^r \left\langle \begin{smallmatrix}W\\ i \end{smallmatrix}\right\rangle \), where \(\left\langle \begin{smallmatrix}W\\ i \end{smallmatrix}\right\rangle \) is the \(W\)-Eulerian number, i.e., the count of elements in \(W\) with descent number equal to \(i\).