resilient codes to phase shift
How can I verify that a code belongs to a group?


Assuming these are really just sub-words that are common to all elements in a group, simply trying out all 8 possible (cyclic) bitshifts of the received sequence to see whether you can match any group representative (e.g. the first column in your GroupN table) would very likely be the most efficient approach – because 8 bit is really not that much, and bit shifts are very efficient on CPUs or on FPGAs (whatever you're using), and even more importantly (since "beacons" surely doesn't sound like high data throughput), the code is easy to understand, and premature optimization is the root of all evil.