Optimality of linear codes over PAM for the modulo-additive Gaussian channel

It has long been known that linear codes achieve the capacity of additive channels over finite fields. Much less has been known about the performance of linear codes over ℤ _M, when used for communication over channels that are additive with respect to this group. Further, linear codes over ℤ _M play an important role in construction of lattices in Euclidean space. When a construction-A lattice is used for communication over an additive white Gaussian noise channel, a modulo-Mℤ additive channel with unimodal noise is induced at the receiver. In this paper it is shown that linear codes over ℤ _M achieve the capacity of such channels when the cardinality of the alphabet is a power of a prime.