This thesis is a study of error-correcting codes for reliable communication in the presence of extreme noise. We consider very noisy channels, which occur in practice by pushing ordinary channels to their physical limits. Both block codes and convolutional codes are examined.
We show that the family of triply orthogonal codes, defined and studied in this thesis, or orthogonal codes can be used to achieve channel capacity for certain classes of very noisy discrete memoryless channels. The performance of binary block codes on the unquantized additive white Gaussian noise channel at very low signal-to-noise ratios is studied. Expressions are derived for the decoder block error as well as bit error probabilities and the asymptotic coding gain near the point where the signal energy is zero.
The average distance spectrum for the ensemble of time-varying convolutional codes is computed, and the result gives a surprisingly accurate prediction of the growth rate of the number of fundamental paths at large distance for fixed codes. A Gilbert-like free distance lower bound is also given. Finally, a Markov chain model is developed to approximate burst error statistics of Viterbi decoding. The model is validated through computer simulations and is compared with the previously proposed geometric model.
