Abelian codes over Galois rings closed under certain permutations

We study n-length Abelian codes over Galois rings with characteristic p^a, where n and p 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 t positions. A code is t-quasi-cyclic (t-QC) if t is an integer such that cyclic shift of a codeword by t 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 t=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 t-QCA code is also an unit-invariant t-QCA code. Unit-invariant Abelian (hence unit-invariant cyclic) and t-QCA codes over Galois field F_p^l and over the integer residue rings are obtainable as special cases.