Databases Reference
In-Depth Information
−
e
1
e
3
−
e
2
e
3
− e
1
e
2
e
1
e
2
e
2
e
3
e
1
e
3
FIGURE C.1
The
A
2
roots embedded in three dimensions.
of the coordinates is always zero. The exact procedure can be found in [
153
,
152
].
E
L
As
we can see from the definition, the
E
L
lattices go up to a maximum dimension of 8. Each of
these lattices can be written as unions of the
A
L
and
D
L
lattices and their translated version.
For example, the
E
8
lattice is the union of the
D
8
lattice and the
D
8
lattice translated by the
vector
1
1
1
1
1
1
1
1
. Therefore, to find the closest
E
8
point to an input
x
, we find
the closest point of
D
8
to
x
, and the closest point of
D
8
to
x
(
2
,
2
,
2
,
2
,
2
,
2
,
2
,
2
)
1
1
1
1
1
1
1
1
−
(
2
,
2
,
2
,
2
,
2
,
2
,
2
,
2
)
, and
pick the one that is closest to
x
.
There are several advantages to using lattices as vector quantizers. There is no need to
store the codebook, and finding the closest lattice point to a given input is a rather simple
operation. However, the quantizer codebook is only a subset of the lattice. How do we know
when we have wandered out of this subset, and what do we do about it? Furthermore, how do
we generate a binary codeword for each of the lattice points that lie within the boundary? The
first problem is easy to solve. Earlier we discussed the selection of a boundary to reduce the
effect of the overload error. We can check the location of the lattice point to see if it is within
this boundary. If not, we are outside the subset. The other questions are more difficult to
resolve. Conway and Sloane [
154
] have developed a technique that functions by first defining
the boundary as one of the quantization regions (expanded many times) of the root lattices.
The technique is not very complicated, but it takes some time to set up, so we will not describe
it here (see [
154
] for details).
We have given a sketchy description of lattice quantizers. For a more detailed tutorial
review, see [
152
].
