Affine invariant extended cyclic codes over Galois rings

Recently, Blackford and Ray-Chaudhuri used transform domain techniques to permutation groups of cyclic codes over Galois rings. They used the same technique to find a set of necessary and sufficient conditions for extended cyclic codes of length 2^m over any subring of GR(4,m) to be affine invariant. Here, we use the same technique to find a set of necessary and sufficient conditions for extended cyclic codes of length p^m over any subring of GR(p^e,m) to be affine invariant, for e=2 with arbitrary p and for p=2 with arbitrary e. These are used to find two new classes of affine invariant Bose-Chaudhuri-Hocquenghem (BCH) and generalized Reed-Muller (GRM) codes over Z₂^e for arbitrary e and a class of affine invariant BCH codes over Z_p² for arbitrary prime p.