Non-Linear Codes on Sparse Graphs

Spatial coupling has been successfully applied to various problems, ranging from error correcting codes to compressive sensing, and shown to have outstanding performance under message-passing algorithms. In particular, spatially coupled low-density parity-check (LDPC) codes were proven to universally achieve capacity under belief-propagation (BP) decoding. In this research proposal, we describe the asymptotic behaviour of spatially coupled LDPC codes, by presenting the results of threshold saturation phenomenon [1]. Then, a finite-length scaling law is presented for protographbased spatially coupled LDPC codes under peeling decoder [2], by analysing the mean evolution of degree-one check nodes. Moreover, we show the connection between the potential functional approach used in [1] and the statistical physics’ replica method [3]. Furthermore, we introduce a class of non-linear checks for both lossy source coding and channel coding on sparse graphs.