How we can encode/decode sparse signals?
Since you have tagged your question compressive-sensing, I would guess that you are interested in a compressive sensing solution to this. In principle, this could work by taking a number (less than 64) of random linear combinations (measurements) of your length 64 vector and transmitting these. At the receiver, the fact that this vector has a sparse representation allows you to reconstruct it again from these measurements. However, compressive sensing does not work that well on low-dimensional vectors (64 is relatively small) and so I do not expect this scheme to work very well.

A more natural solution, since you mention that you know the sparse length 128 representation of the vector, is to use run-length encoding (Wikipedia explanation) of the sparse representation and transmit that. Since the sparse vector is almost all zeros, this should be very efficient.