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

Berger and Charpin (see ibid., vol.42, p.2194-2209, 1996 and Des., Codes Cuyptogr., vol.18, no.1/3, p.29-53, 1999) devised a theoretical method of calculating the permutation group of a primitive cyclic code over a finite field using permutation polynomials and a transform description of such codes. We extend this method to cyclic and extended cyclic codes over the Galois ring GR (p^a, m), developing a generalization of the Mattson-Solomon polynomial. In particular, we classify all affine-invariant codes of length 2^m over Z₄ , thus generalizing the corresponding result of Kasami, Lin, and Peterson (1967) and giving an alternative proof to Abdukhalikov. We give a large class of codes over Z₄ with large permutation groups, which include generalizations of Bose-Chaudhuri-Hocquenghem (BCH) and Reed-Muller (RM) codes