On complexity of trellis structure of linear block codes

This paper is concerned with the trellis structure of linear block codes. The paper consists of four parts. In the first part, we investigate the state and branch complexities of a trellis diagram for a linear block code. A trellis diagram with the minimum number of states is said to be minimal. First, we express the branch complexity of a minimal trellis diagram for a linear block code in terms of the dimensions of specific subcodes of the given code. Then we derive upper and lower bounds on the number of states of a minimal trellis diagram for a linear block code, and show that a cyclic(or shortened cyclic) code is the worst in terms of the state complexity among the linear block codes of the same length and dimension. Furthermore, we show that the structural complexity of a minimal trellis diagram for a linear block code depends on the order of its bit positions. This fact suggests that an appropriate permutation of the bit positions of a code may result in an equivalent code with a much simpler minimal trellis diagram. In part two, we consider boolean polynomial representation of codewords of a linear block code. This representation will help us in study of the trellis structure of the code. In part three, we apply boolean polynomial representation of a code to construct its minimal trellis diagram. Particularly, we focus on the construction of minimal trellises for Reed-Muller codes and the extended and permuted binary primitive BCH codes which contain Reed-Muller code as subcodes. Finally, we an_yze and present the structural complexity of minimal trellises for the extended and permuted (64,24), (64,45), and double-error-correcting (2", 2"-2m1) BCH codes. We show that these codes have relatively simple trellis structure and hence can be decoded with the Viterbi decoding algorithm.