  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.
