A transform approach to permutation groups of cyclic codes over Galois rings

Recently, Berger and Charpin [3], [4] devised a theoretical method of calcu lating the permutation group of a primitive cyclic code over a finite field using permutation polynomials and a transform description of such codes. W e extend this method to cyclic and extended cyclic codes over the Galois ri ng GR(p(a), m), developing a generalization of the Mattson-Solomon polynomi al. Tn particular, we classify all affine-invariant codes of length 2(m) ov er Z(4), thus generalizing the corresponding result of Kasami, Lin, and Pet erson [9] and giving an alternative proof to Abdukhalikov [1], me give a la rge class of codes over Z(4) with large permutation groups, which include g eneralizations of Bose-Chaudhuri-Hocquenghem (BCH) and Reed-Muller (RM) cod es.