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.

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.