Optimal Lattice Codes For the Gaussian Channel

Abstract

<p>Lattices are used to construct a class of equal-energy codes for the Gaussian channel and the resultant error probability tends to zero for large n at all rates below channel capacity. The error probability is explicitly bounded for any given lattice code and then further further bounded for a general code using the Minkowski-Hlawka theorem of the geometry of numbers. Similar bounds are applied also to maximum-energy codes, to show that such lattice codes are near-optimal.</p> <p>Finally, the error bounds are applied to explicit codes defined for all n=2ᵐ. These codes are shown to have a low Pℯ at rates higher than any previously attained.</p>