**Efficient quantum communication under collective noise**

We introduce a new quantum communication protocol for the transmission of quantum information under collective noise. Our protocol utilizes a decoherence-free subspace in such a way that an optimal asymptotic transmission rate is achieved, while at the same time encoding and decoding operations can be efficiently implemented. The encoding and decoding circuit requires a number of elementary gates that scale linearly with the number of transmitted qudits, $m$. The logical depth of our encoding and decoding operations is constant and depends only on the channel in question. For channels described by an arbitrary discrete group $G$, i.e.~with a discrete number, $\lvert G\rvert$, of possible noise operators, perfect transmission at a rate $m/(m+r)$ is achieved with an overhead that scales at most as $\mathcal{O}(d^r)$ where the number of auxiliary qudits, $r$, solely depends on the group in question. Moreover, this overhead is independent of the number of transmitted qudits, $m$. For certain groups, e.g. cyclic groups, we find that the overhead scales only linearly with the number of group elements $|G|$. For continuous groups we devise an efficient scheme for approximate transmission, and examine in detail the case of collective phase noise channels described by the group $U(1)$. The same scheme can also be utilized for the storage of quantum information in the presence of collective noise.