  We study noiseless subsystems on collective rotation channels of qudits,
i.e., quantum channels with operators in the set ${\mathcal E}(d,n) = \{
U^{\otimes n}: U \in {\mathrm{SU}}(d)\}.$ This is done by analyzing the
decomposition of the algebra ${\mathcal A}(d,n)$ generated by ${\mathcal
E}(d,n)$. We summarize the results for the channels on qubits ($d=2$), and
obtain the maximum dimension of the noiseless subsystem that can be used as the
quantum error correction code for the channel. Then we extend our results to
general $d$. In particular, it is shown that the code rate, i.e., the number of
protected qudits over the number of physical qudits, always approaches 1 for a
suitable noiseless subsystem. Moreover, one can determine the maximum dimension
of the noiseless subsystem by solving a non-trivial discrete optimization
problem. The maximum dimension of the noiseless subsystem for $d = 3$ (qutrits)
is explicitly determined by a combination of mathematical analysis and the
symbolic software Mathematica.
