Abelian codes over Galois rings closed under certain permutations

We study -length Abelian codes over Galois rings with characteristic , where and are relatively prime, having the additional structure of being closed under the following two permutations: i) permutation effected by multiplying the coordinates with a unit in the appropriate mixed-radix representation of the coordinate positions and ii) shifting the coordinates by positions. A code is -quasi-cyclic ( -QC) if is an integer such that cyclic shift of a codeword by positions gives another codeword. We call the Abelian codes closed under the first permutation as unit-invariant Abelian codes and those closed under the second as quasi-cyclic Abelian (QCA) codes. Using a generalized discrete Fourier transform (GDFT) defined over an appropriate extension of the Galois ring, we show that unit-invariant Abelian and QCA codes can be easily characterized in the transform domain. For = 1, QCA codes coincide with those that are cyclic as well as Abelian. The number of such codes for a specified size and length is obtained and we also show that the dual of an unit-invariant -QCA code is also an unit-invariant -QCA code. Unit-invariant Abelian (hence unit-invariant cyclic) and -QCA codes over Galois field and over the integer residue rings are obtainable as special cases.