Optimal lattice constant

In the previous subsection, the coding region was quantised with hypercubes. In general this is optimal only in the single-dimensional case. For example, in 2-dimensions, the coding region should be quantised with hexagons instead. This is because hexagons tessellate more tightly than squares. In higher dimensions, more complex polytopes would be the ideal quantisation primitive [#!Farr:1999!#]. Note that a hyper-sphere cannot be used as it fails to tessellate without overlap or gaps. This would mean that the decoder cannot uniquely decode the message because a model may correspond to multiple coding region or to none.

More detailed work upon this problem of dense-packing has been done by Conway and Sloane [#!Conway.Sloane:1982!#,#!Conway.Sloane:1988!#]. However, for the purposes for determining the optimal message length, it is sufficient to know that the message length equation is altered to -

$$\textrm{MesgLen~} = -\log_e \frac{h(\vec{\theta})f(\vec{x}\v... ...heta})}{\sqrt{F(\vec{\theta})}} + \frac{n}{2} (1 + \log_e k_n)$$ (18)

where k_n is the n-dimensional lattice constants [#!Farr:1999!#].

n	kn
1	$1/12 \approx 0.0833$
2	$5/36\sqrt{3} \approx 0.0802$
...	...
$\infty$	$1/2\pi e \approx 0.0585$