Let U be a region of L-dimensional Euclidean space, and let UK be a region that

Let U be a region of L-dimensional Euclidean space, and let UK be a region that is the K-fold Cartesian product of U. We can also show that the shaping gain of a Cartesian-product region SK is In words, the shaping gain of cK is equal to the shaping gain of C. This result is fundamental to the understanding of multidimensional constellations (and more generally signal-space coding). Taking a Cartesian product of lattice codes does not affect the coding or the shaping gain. The way to achieve an increase in coding and shaping gains as the dimensionality is increased is to choose the components of the code dependently.

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