Hard and soft decision decoding of phase rotation invariant block codes

We consider a binary discrete channel with phase inversion and with random errors. Such a channel is a cascade of two channels. We also consider a continuous channel with phase inversion. For both channels, we wish to correct a phase inversion error when it occurs as well as random errors. For this purpose, we introduce a special metric to describe combinations of these errors. We call this metric the phase rotating metric (PR-metric). In this paper, we consider constructions of PR-metric codes together with their hard decision decoding for a discrete channel and their soft decision decoding for a continuous channel with phase inversion.