A linear block code has a finite-length trellis diagram which terminates at the length of the code. Such a trellis diagram is expressed and constructed in terms of sections. The complexity of an L-section trellis diagram for a linear block code is measured by the state and branch complexities, the state connectivity, and the parallel structure. The state complexity is defined as the number of states at the end of each section of the L-section trellis diagram, the branch complexity is defined as the number of parallel branches between two adjacent states, the state connectivity is defined in terms of the number of states which are adjacent to and from a given state and the connections between disjoint subsets of states, and the parallel structure is expressed in terms of the number of parallel sub-trellis diagrams without cross connections between them. This paper investigates the branch complexity, the state connectivity, and the parallel structure of an L-section minimal trellis diagram for a binary linear block code. First these complexities and structure are expressed in terms of the dimensions of specific subcodes of the given code. Then, the complexities of 2i-section minimal trellis diagrams for Reed-Muller codes are given.

The performance evaluation by analysis of systems employing Reduced State Viterbi decoding is addressed. This type of decoding is characterized by an inherent error propagation effect, which yields a difficulty in the error probability analysis, and has been usually neglected in the literature. By modifying the Full State trellis diagram, we derive for Reduced State schemes, new transfer function bounds with the effects of error propagation. Both the Chernoff and the tight upper bound are applied to the transfer function in order to obtain the bit error probability upper bound. Furthermore, and in order to get a tighter bound for Reduced State decoding schemes with parallel transitions, the pairwise probability of the two sequences involved in an error event is upper bounded, and then the branch metric of a sequence taken from that bound is associated with a truncated instead of complete Gaussian noise probability density function. To support the analysis, particular assessment is done for a Trellis Coded Modulation scheme.

In the maximum-likelihood decoding under a non-Gaussian noise, the decoding region is bounded by complex curves instead of a perpendicular bisector corresponding to the Gaussian noise. Therefore, the error rate is not evaluated by the Euclidean distance. The Bhattacharyya distance is adopted since it can evaluate the error performance for a noise with an arbitrary distribution. Upper bound formulae of a bit error rate and an event error rate are obtained based on the error-weight-profile method proposed by Zehavi and Wolf. The method is modified for a non-Gaussian channel by using the Bhattacharyya distance instead of the Euclidean distance. To determine the optimum code for an impulsive noise channel, the upper bound of the bit error rate is calculated for each code having an encoder with given shift-register lehgth. The best code is selected as that having the minimum upper bound of the bit error rate. This method needs much computation time especially for a code with a long shift-register. To lighten the computation burden, a suboptimum search is also attempted. For an impulsive noise, modeled from an observation in digital subscriber loops, an optimum or suboptimum code is searched for among codes having encoders with a shift-register of up to 4 bits. By using a code with a 4-bit encoder, a coding gain of 20 dB is obtained at the bit error rate 10-5. It is 11 dB more than that obtained by Ungerboeck's code.