Towards a Theory of Codes for Iterative Decoding

Abstract: Channel codes in combination with iterative decoding techniques are a both powerful and efficient method to protect data against disturbances in digital communication systems. This thesis deals with various code constructions for block-wise and continuous transmission that have the potential to achieve low bit error rates with iterative decoding, even when operating close to the Shannon limit. For a performance analysis of iterative decoding schemes with convolutional component codes it is of great value to understand and describe the exact behavior of a convolutional code. Analytic expressions for the bit error probabilities corresponding to maximum likelihood decoding of a rate one-half, memory two convolutional code are derived as a function of the crossover probability of the binary symmetric channel. Three different encoders for the same code are considered, and the Viterbi decoder is compared to the Max-Log-MAP decoder, which is often used in iterative decoding schemes as an alternative to an optimal a posteriori probability (APP) decoder. An ensemble of self-concatenated convolutional codes is introduced and the average weight spectrum of the codes is expressed by the solution to a system of recurrent equations. Based on this average weight spectrum a lower bound on the free distance and upper bounds on the burst error probability for maximum likelihood decoding are derived. A substantial part of the thesis is devoted to proving for various code families the existence of and deriving bounds on the iterative limit. This threshold determines when arbitrarily low error probabilities can asymptotically be achieved with iterative decoding. Braided block codes, a new code construction for continuous transmission based on block component codes, are introduced. For braided block codes with Hamming component codes as well as for the block and convolutional versions of low-density parity-check codes and turbo codes, bounds on the iterative limits are derived and compared to results of computer simulations. Furthermore, it is shown with a constructive permutor design procedure that there exist turbo codes for which the minimum distance grows logarithmically with the block length.