The structure of the twisted squaring construction, a generalization of the squaring construction, is studied with respect to trellis diagrams and complexity. We show that binary affine-invariant codes, which include the extended primitive BCH codes, and the extended binary quadratic-residue codes, are equivalent to twisted squaring construction codes. In particular, a recursive symmetric reversible design of the BCH codes is derived. Using these constructions, the parameters of the minimal trellis diagram of the BCH codes are determined, including the componentwise state-space profile and trellis complexity. New designs and permutations that yield low trellis complexity for the quadratic-residue codes are presented. Generalized Hamming weights are derived from these constructions. As an example, the (48, 24, 12) quadratic-residue code is analyzed, a strictly componentwise optimal permutation is derived, and the corresponding state-space profile and complete generalized Hamming weight hierarchy are obtained.

The trellis complexity of composite-length cyclic codes (CLCC's) is addressed. We first investigate the trellis properties of concatenated and product codes in general. Known factoring of CLCC's into concatenated subcodes is thereby employed to derive upper bounds on the minimal trellis size and state-space profile. New decomposition of CLCC's into product subcodes is established and utilized to derive further upper hounds on the trellis parameters. The coordinate permutations that correspond to these bounds are exhibited. Additionally, new results on the generalized Hamming weights of CLCC's are obtained. The reduction in trellis complexity of many CLCC's leads to soft-decision decoders with relatively low complexity.>

A systematic design of a trellis-based maximum-likelihood soft-decision decoder for linear block codes is presented. The essence of the decoder is to apply an efficient search algorithm for the error pattern on a reduced trellis representation of a certain coset. Rather than other efficient decoding algorithms, the proposed decoder is systematically designed for long codes, as well as for short codes. Computational gain of up to 6 is achieved for long high-rate codes over the well-known trellis decoder of Wolf (1978). Efficient decoders are also obtained for short and moderate length codes.>

The size of minimal trellis representation of linear block codes is addressed. Two general upper bounds on the trellis size, based on the zero-concurring codewords and the contraction index of the subcodes, are presented. The related permutations for attaining the bounds are exhibited. These bounds evidently improve the previously published general bound. Additional bounds based on certain code constructions are derived. The focus is on the squaring construction, and specific constructive bounds for Reed-Muller and repeated-root cyclic codes are obtained. In particular, the recursive squaring construction of Reed-Muller codes is explored and the exact minimal trellis size of this design is obtained. Efficient permutations, in the sense of the trellis size, are also demonstrated by using shortening and puncturing methods. The corresponding bounds are specified.>