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Most of the basic concepts of algebraic coding theory are derived for the memoryless binary symmetric channel. These concepts do not necessarily hold for time-varying channels or for channels with memory. However, errors in real-life channels seem to occur in bursts rather than independently, suggesting that these channels exhibit some statistical dependence or memory. Nonetheless, the same algebraic codes are still commonly used in current communication systems that employ interleaving to spread channel error bursts over the set of received codewords to make the channel appear memoryless to the block decoder. This method suffers from immediate shortcomings as it fails to exploit the channel’s memory while adding delay to the system. We study optimal maximum likelihood block decoding of binary codes sent over several binary additive channels with infinite and finite memory. We derive conditions on general binary codes and channels parameters under which maximum likelihood and minimum distance decoding are equivalent. The channels considered in this work are the infinite and finite memory Polya contagion channels, the queue-based channel, and the Gilbert-Elliott channel. We also present results on the optimality of classical perfect and quasi-perfect codes when used over the aforementioned channels under maximum likelihood decoding. / Thesis (Master, Mathematics & Statistics) -- Queen's University, 2013-07-12 13:45:35.294

The aim of' this investigation is to determine the combination of signal coding and modulation for satellite links, that, for a given degree of equipment complexity needed for the detection of the received signal, achieves the best tolerance to noise. Computer simulation tests and theoretical analyses are used to compare the various proposed signal designs The trellis coded M-ary phase-shift-keyed (MPSK) modulation method is introduced as the scheme for which different codes are to be devised. A class of known binary convolutional codes for 8 and 16 PSK signals is studied, and new correlative-level codes using modulo-M arthimetic are designed for MPSK signals. The soft-decision maximum likelihood Viterbi decoding algorithm is considered for the two proposed signal designs, and a more conventional near-maximum likelihood (reduced-state Viterbi) decoding scheme is also investigated for both types of coded signals. Two novel decoding schemes, derived from a more conventional near-maximum likelihood decoder, are proposed for coded 8PSK signals. In both decoders the amount of computation involved in decoding each data-symbol is adjusted to meet the prevailing noise level in transmission. Results of extensive computer simulation tests for both decoding schemes are presented. These results suggest that the new schemes come very close to achieving the maximum likelihood decoding of the coded signals without, however, requiring nearly as much storage and computation per decoded data symbol as does the Viterbi decoder. The carrier-phase synchronisation prob1em in a coherent trellis coded MPSK system is investigated. Eight new rotationally invariant rate-2/3 and rate-3/U convolutional codes for 8 and 16 PSK signals are designed. The new coded MPSK signals, when combined with a simple phase-error correction system proposed for the receiver, are able to tolerate the likely carrier-phase changes in the reference carriers of the coherent demodulation process and therefore avoid the prolonged error bursts that are otherwise caused in the decoded data symbols by such phase shifts. coding gains of the majority of the new codes The asymptotic here are either the same as, or come close to, those of the best known but not rotationally invariant convolutional codes of the same rates.

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Rate 1/2 systematic recursive convolutional codes over integer rings modulo-q are investigated for their performance. The investigation examines the performance in severe fading and additive white Gaussian noise for codes with various constraint lengths. The arithmetic for the codes is modulo-q. where the value of q is within the range of 2 to 16. An exhaustive search is carried out for codes with short constraint lengths. A reduced search is developed for larger constraint lengths which restricts the tap polynomials to irreducible polynomials over Zq. The irreducible polynomials are generated and the ones not found in the literature are presented in tables. The search algorithms are outlined and the results for the codes are tabulated. The performance of selected codes are verified by Monte-Carlo simulation techniques. Several codes have better performance than comparable codes presented in the literature for the Rayleigh fading channel. In sme of cases, the codes found have better performance on the AWGN channel than the best known ring codes. The characteristics of rotationally invariant (RI) ring codes presented in the literature are used in an exhaustive search for codes over Zq which are invariant to phase shifts of 2[pi]/q. Tables of RI codes optimized for the Rayleigh fading channel are presented along with codes which are optimized for the AWGN channel. / Graduate