On Cyclic Polar Codes and The Burst Erasure Performance of Spatially-Coupled LDPC Codes

Abstract

Polar codes were introduced in 2009 and proven to achieve the symmetric capacity of any binary-input discrete memoryless channel under low-complexity successive cancellation decoding. In this thesis, we construct cyclic polar codes based on a mixed-radix Cooley-Tukey decomposition of the Galois field Fourier transform. The main results are: we can, for the first time, construct, encode and decode polar codes that are cyclic, with their blocklength being arbitrary; for a given target block erasure rate, we can achieve significantly higher code rates on the erasure channel than the original polar codes, at comparable blocklengths; on the symmetric channel with only errors, we can perform much better than equivalent rate Reed-Solomon codes with the same blocklength, by using soft-decision decoding; and, since the codes are subcodes of higher rate RS codes, a RS decoder can be used if suboptimal performance suffices for the application as a trade-off for higher decoding speed. In 2010, it was shown that spatially-coupled low-density parity-check (LDPC) codes approach the capacity of binary memoryless channels, asymptotically, with belief-propagation (BP) decoding. In this thesis, we are interested in the finite length average performance of randomly coupled LDPC ensembles on binary erasure channels with memory. The significant contributions of this work are: tight lower bounds for the block erasure probability ($P_B$) under various scenarios for the burst pattern; bounds focused on practical scenarios where a burst affects exactly one of the coupled codes; expected error floor for the bit erasure probability ($P_b$) on the binary erasure channel; and, characterization of the performance of random regular ensembles, on erasure channels, with a single vector describing distinct types of size-$2$ stopping sets. All these results are verified using Monte-Carlo simulations.