THE TWISTED SQUARING CONSTRUCTION, TRELLIS COMPLEXITY, AND GENERALIZED WEIGHTS OF BCH AND QR CODES

The structure of the twisted squaring construction, a generalization o f the squaring construction, is studied with respect to trellis diagra ms and complexity, We show that binary affine-invariant codes, which i nclude the extended primitive BCH codes, and the extended binary quadr atic-residue codes, are equivalent to twisted squaring construction co des. In particular, a recursive symmetric reversible design of the BCH codes is derived, Using these constructions, the parameters of the mi nimal trellis diagram of the BCH codes are determined, including the c omponentwise state-space profile and trellis complexity, New designs a nd permutations that yield low trellis complexity for the quadratic-re sidue codes are presented, Generalized Hamming weights are derived fro m these constructions, As an example, the (48, 24, 12) quadratic-resid ue code is analyzed, a strictly componentwise optimal permutation is d erived, and the corresponding state-space profile and complete general ized]Clamming weight hierarchy are obtained.