I'm currently working on QRAC and was wondering if there's an encoding protocol in $3 \rightarrow 1$ such that the receiver is able to retrieve any one of the XOR combinations of the bits, along with the original bits itself ( a grand total of 7 functions; if a, b, c are the bit positions, the receiver should be able to guess a, b, c, a+b, b+c, c+a, a+b+c with a probability greater than 1/2).
Is this even possible? My initial guess would be to use some kind of POVM measurements in the decoding part but don't how to proceed with that.
If you have any ideas on this then please let me know. Thank you.