> However, Reed–Solomon codes have the disadvantage that they are practical only for small K, N, and q: standard implementations of encoding and decoding have a cost of order K(N-K)log2 N packet operations. Furthermore, with a Reed–Solomon code, as with any block code, one must estimate the erasure probability f and choose the code rate R=K/N before transmission. If we are unlucky and f is larger than expected and the receiver receives fewer than K symbols, what are we to do? We would like a simple way to extend the code on the fly to create a lower-rate (N', K) code. For Reed–Solomon codes, no such on-the-fly method exists.
You don't have to choose the rate before transmission. Reed-Solomon code is equivalent to sampling a K-degree polynomial. You can sample it all you want, i.e. increase N on-the-fly, up to the number of the elements in your field, if finite. Say, you want to broadcast your K blocks to L recipients reliably: after you have sent the K blocks you start sampling the polynomial and sending the redundant blocks. You don't need to decide a priory how many redundant blocks you will send: as soon as the last recipient reports receiving K blocks you are done. GF(2^64) is as good as infinite in that scenario.