Decoding Binary Linear Codes Over Channels With Synchronization Errors

Time synchronization is crucial for the safe and reliable operation of the fifth generation (5G) network, especially for applications requiring ultra-reliable low-latency data transmissions. The time synchronization problem, however, becomes increasingly challenging in high-mobility scenarios because the channel conditions, e.g., the multipath delay spread may vary rapidly. While there exist numerous works on the design of efficient channel decoding algorithms, decoding linear codes such as polar codes in the presence of synchronization errors is a less-explored topic. In this paper, we aim to fill this void and develop a systemic approach to decode general binary linear codes over binary symmetric channels with synchronization errors in which the lack of synchronization is modeled as the deletion channel model. The maximum likelihood (ML) decoding problem for binary linear codes over deletion channels is first formulated as a nonlinear optimization problem, in which a set of linear constraints are employed to characterize the input-output relationship of a deletion channel. It turns out that both the objective function and the constraints of this optimization problem are nonlinear, which poses significant challenges against the design of efficient decoding algorithms. As a remedy, we first replace the nonlinear objective function of this optimization problem via a lower bound. And we prove this lower bound is a linear function in the special case that the input is binary. We then apply the linear programming (LP) relaxation approach to obtain an approximate solution to the proposed nonlinear optimization problem. An adaptive branch-and-cut decoding algorithm has also been developed by making use of the ML-certificate property of the LP decoder for deletion channel. It is seen through simulation studies that the proposed decoding algorithm can achieve close-to-optimal bit error rate (BER) decoding performance at moderate computational complexity.