The Affine Groups

Abstract

Although the extended quadratic residue codes are the only codes which are known to be invariant under the projective unimodular subgroup of the linear fractional group, many cyclic codes are known which have extensions that are invariant under a different transitive group of permutations. The length of each q-ary cyclic code whose extension is invariant under this type of group is of the form n = q^m− 1. The coordinates of the cyclic code are associated with the nonzero symbols in GF(q^m). The extended code is invariant under the cyclic shift, which fixes 0 and cycles the nonzero coordinates according to the permutation C_u → C_v, v = αu in GF(q^m). The group of all cyclic shifts is just the multiplicative group of GF(q^m), C_u → C_v, v = βu, where u,v ∈ GF(q^m) and β ∈ GF(q^m) − {0}. The extended cyclic code may also be invariant under the additive group in GF(q^m), [this group is more conveniently called the translational group], C_u → C_v, v = u + β, v, u, β ∈ GF(q^m). If the extended cyclic code is invariant under both cyclic shifts and translations, it is invariant under the group generated by these permutations.