Encoding and decoding algorithms for an optimal lattice-based code

Shannon's random coding theorem gives an existence proof that codes exist which can transmit at rates arbitrarily close to channel capacity. However, no construction for such a random code has been given. This paper describes an optimal code which is not random, but has lattice structure. The code is constructed by selecting 2 to the 14th power points from a hypersphere in an 8-dimensional spherepacking lattice. Encoding and decoding strategies for this lattice code are described. The high degree of symmetry in the lattice code is partially exploited to produce encoding and decoding algorithms, which are unexpectedly simple. At a bit-error-rate of 0.00001, this lattice code is 5 dB better than 12 phase PSK and 2 dB better than 4 level QAM, both of which require about the same bandwidth as the lattice code.