  We give a full explanation of the noiseless subsystem that protects a
single-qubit against collective errors and the corresponding recursive scheme
described by C.-K. Li et. al. [Phys. Rev. A 84, 044301 (2011)] from a
representation theory point of view. Furthermore, we extend the construction to
qudits under the influence of collective SU($d$) errors. We find that under
this recursive scheme, the asymptotic encoding rate is $1/d$.
