Asynchronous compute-and-forward/integer-Forcing with quasi-cyclic codes

Communication in the presence of bounded timing asynchronism which is known to the receiver but cannot be easily compensated is studied. Examples of such situations include point-to-point communication over inter-symbol interference (ISI) channels and asynchronous wireless networks. In these scenarios, although the receiver may know all the delays, it may not be an easy task for the receiver to compensate the delays as the signals are mixed together. A novel framework called interleave/deinterleave transform (IDT) is proposed to deal with this problem. It is shown that the IDT allows one to design the delays so that quasi-cyclic (QC) codes with a proper shifting constraint can be used accordingly. When used in conjunction with QC codes, IDT provides significantly better performance than existing schemes relying solely on cyclic codes. Two instances of asynchronous physical-layer network coding, namely the integer-forcing equalization for ISI channels and asynchronous compute-and-forward, are then studied where the gap-to-capacity can be bridged for the former and significant gains can be obtained for the later. The proposed IDT can be thought of as a generalization of the interleaving/deinterleaving idea in [1] which allows the use of QC codes thereby substantially increasing the design space.