Geometric Relationships Between Gaussian and Modulo-Lattice Error Exponents

Lattice coding and decoding have been shown to achieve the capacity of the additive white Gaussian noise (AWGN) channel. This was accomplished using a minimum mean-square error scaling and randomization to transform the AWGN channel into a modulo-lattice additive noise channel of the same capacity. It has been further shown that when operating at rates below capacity but above the critical rate of the channel, there exists a rate-dependent scaling such that the associated modulo-lattice channel attains the error exponent of the AWGN channel. A geometric explanation for this result is developed. In particular, it is shown how the geometry of typical error events for the modulo-lattice channel coincides with that of a spherical code for the AWGN channel.