Balanced codes and nonequiprobable signaling

The problem of shaping signal constellations that are designed for the Gaussian channel is considered. The signal constellation consists of all points from some translate of a lattice Λ that lie within a region R. The signal constellation is partitioned into T annular subconstellations Ω_o,…,Ω_T-1, by scaling the region R. Signal points in the same subconstellation are used equiprobably, and a shaping code selects region Ω_i with frequency f_i. If the signal constellation is partitioned into annular subconstellations of unequal size. then the transmission rate should vary with the choice of codeword in the shaping code. and it will be necessary to queue the data in buffers. It is described how the balanced binary codes constructed by D. E. Knuth (1986) can be used to avoid a data rate that is probabilistic. The basic idea is that if symbols 0 and 1 represent constellations of unequal size. and if all shaping codewords have equally many 0´s and 1´s, then the data rate will be deterministic