Reflecting on the AWGN error exponent

Recently it was shown that a lattice code with lattice decoding can achieve the capacity of the additive white Gaussian noise (AWGN) channel. This was achieved by using a minimum mean-square error (MMSE) scaling and dithering to transform the AWGN channel into a modulo-lattice additive noise (mod-Λ) channel. Further, Liu et. al. have shown that lattice decoding can achieve the error exponent of the AWGN channel using a scaling other than the MMSE scaling at rates above the critical rate of the channel. We present a simple geometric explanation for this result.