Automorphism groups of cyclic codes over Z_4 and other local rings

摘要

The recent discovery that good binary nonlinear codes can be constructed from Z_4-linear codes via a Gray map has motivated the study of codes over rings. In particular, much has been written about cyclic codes over the rings Z_4 and F_2 + uF_2. A discrete Fourier transform makes cyclic codes over certain local rings (such as Z_(pa)) easy to classify. It also facilitates the computation of permutation groups of cyclic and affine-invariant codes over these rings. We give the necessary and sufficient conditions for affine-invariant codes to exist over Z_4, and we give some with a particularly large automorphism group. We also show how to compute the permutation groups of some binary repeated-root cyclic codes using a similar approach, by viewing them as coming from con-stacyclic codes in F_2 + uF_2.