  We consider entanglement purification protocols for multiple copies of qubit
states. We use high-dimensional auxiliary entangled systems to learn about
number and positions of errors in the noisy ensemble in an explicit and
controlled way, thereby reducing the amount of noise in the ensemble and
purifying the remaining states. This allows us to design entanglement
purification protocols for any number of copies that work particularly well for
a small number of expected errors, i.e. high fidelity of initial states. The
main tool is a counter gate with which the required non-local information can
be transferred into the high-dimensional entangled qudit auxiliary states. We
compare our schemes to standard recurrence protocols that operate on pairs of
copies, and hashing and breeding protocols that operate on a (asymptotically)
large number of copies. Our protocols interpolate between these two regimes,
leading to a higher achievable fidelity and yield. We illustrate our approach
for bipartite qubit states, and generalize it to purify multi-party GHZ states.
