Some optimal codes have structure

The techniques of the geometry of numbers, especially the Minkowski-Hlawka theorem, are used to modify Shannon´s existence proof for optimal channel codes, so that the modified proof applies specifically to lattice codes. The resulting existence proof states that there exist lattice codes which satisfy Shannon´s bound to within the factor 4, and hence match the reliability exponent and critical rate bounds which Shannon derived for optimal codes with unspecified structure. Therefore, it is demonstrated that optimal codes need not be random, but rather that some of them have structure, e.g. the structure of a lattice code