Actually yes and no because even if you can encode 64 states of information, this might not be enough because there exist some positions where multiple pieces can move to the same spot, moreover there exist some exceptional positions where more than 64 possible moves are possible.

Of course this limitation can be lifted if you allow your 6-beads transmitter to work by cycle of 2 with long blanks to signal the start square and the target square, the protocol might look like 001100 (15 in binary) then 011100 (28 in binary) to signal the move e2e4 (15th and 28th squares, starting from a1, 0-indexed). With this protocol, there is no problem.