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In this paper we show that any sequence of infinite lattice constellationswhich is good for the unconstrained Gaussian channel can be shaped into acapacity-achieving sequence of codes for the power-constrained Gaussian channelunder lattice decoding and non-uniform signalling. Unlike previous results inthe literature, our scheme holds with no extra condition on the lattices (e.g.quantization-goodness or vanishing flatness factor), thus establishing a directimplication between AWGN-goodness, in the sense of Poltyrev, andcapacity-achieving codes. Our analysis uses properties of the discrete Gaussiandistribution in order to obtain precise bounds on the probability of error andachievable rates. In particular, we obtain a simple characterization of thefinite-blocklength behavior of the scheme, showing that it approaches theoptimal dispersion coefficient for \textit{high} signal-to-noise ratio. Wefurther show that for \textit{low} signal-to-noise ratio the discrete Gaussianover centered lattice constellations can not achieve capacity, and thus a shift(or "dither") is essentially necessary.

AWGN-Goodness is Enough: Capacity-Achieving Lattice Codes based on Dithered Probabilistic Shaping