The twisted squaring construction, trellis complexity, and generalized weights of BCH and QR codes

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.