Chain Codes and Spherical Tits Buildings

To a $n$-dimensional vector space $V$ over a finite field $\mathbb{F}_q$ it is possible to associate a structure of spherical Tits building. The chambers of such building are maximal flags: maximal sequences of nested subspaces. In the case $q=2$, there is a unique $( n-1) $-dimensional 1-MDS code $C\subset V$. We show the existence of chambers associated to such a code that are chain type (in the sense of codes theory) and given a complete characterization of the connected components of the chain type chambers.