Combined coding and shaping over a multitone channel

The idea of combined shaping and coding of a signal constellation over a multitone channel is introduced. Two joint optimization methods, partly integer, for distributing the rate and the energy are introduced. In the first method, the minimum distance to noise ratio (protection) along all the dimensions is the same. The proposed method maximizes this protection. In the second method, this restriction is relaxed. The second method outperforms the first. As part of the calculations, a closed-form formula has been found for the weight distribution of the scaled E₈ lattice