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We present a new lower bound for the error exponents of nested lattice codes for the additive white Gaussian noise (AWGN) channel. The exponents are closely related to those of an unconstrained additive noise channel where the noise is a weighted sum of a white Gaussian and a spherically uniform random vector. The new lower bound improves the previous result derived by Erez and Zamir (2002) and stated in terms of the Poltyrev exponents. More surprisingly, the new lower bound coincides with the random coding error exponents of the optimal Gaussian codes for the AWGN channel in the nonexpurgated regime. One implication of this result is that minimum mean squared error (MMSE) scaling, despite its key role in achieving capacity of the AWGN channel, is no longer fundamental in achieving the best error exponents for rates below channel capacity. These exponents are achieved using a lattice inflation parameter derived from a large-deviation analysis.