This paper generalizes the results from Wolfmann (see ibid., vol.45, p.2527-2532, Nov. 1999 and vol.47, p.1773-1779, July 2001), classifying all negacyclic codes over Z/sub 4/ of even length using a transform approach. It is then shown which linear binary cyclic codes are images of negacyclic codes under the Gray map. In the process, the concatenated structure of both negacyclic codes and binary repeated-root cyclic codes is given.

We characterize codes over Z/sub p/ which are the Gray images of (1-p/sup k/)-cyclic codes or cyclic codes over Z(p/sup k+l/) (k/spl ges/1). A necessary and sufficient condition for the Gray image of a Z(p/sup 2/)-linear (1-p)-cyclic code to be linear is given. In many cases, this yields an explicit description of the Gray image of a linear (1-p)-cyclic code over Z(p/sup 2/), of length relatively prime to p. Linear cyclic codes over Z(p/sup 2/) whose Gray images are linear cyclic codes over Z/sub p/ have been characterized. Some generalizations of these results to the case of Z(p/sup k+1/), where k/spl ges/2, are also obtained.

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/sup a/, m), developing a generalization of the Mattson-Solomon polynomial. In particular, we classify all affine-invariant codes of length 2/sup m/ over Z/sub 4/, 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 Z/sub 4/ with large permutation groups, which include generalizations of Bose-Chaudhuri-Hocquenghem (BCH) and Reed-Muller (RM) codes.