[throw5away]

You don't need the n bits of information from each ballot. You can just transmit the aggregate counts. A human can easily put tally marks in each of n buckets. In fact, a computer would likely do the same instead of creating an exponential number of buckets...

If you insist on a degenerate formulation, just split the ballots into n different elections and transmit the results of each one of those elections. We can use the same equipment we have today.

[alisonkisk]

Even simpler, an approval voter can cast up to N (different) ballots.

[dane-pgp]

I like this, although with paper ballots it would mean counting (up to) N times as many pieces of paper, which would be (up to) N times slower/more expensive.