Databases Reference
In-Depth Information
The Root Lattices
Define
e
i
to be a vector in
L
dimensions whose
i
th component is 1 and all other
components are 0. Some of the root systems that are used in lattice vector
quantization are given as follows:
e
i
e
j
D
L
±
±
,
i
=
j
,
i
,
j
=
1
,
2
,...,
L
e
L
+
1
i
e
L
+
1
j
A
L
±
(
−
)
,
i
=
j
,
i
,
j
=
1
,
2
,...,
L
e
i
e
j
E
L
±
±
,
i
=
j
,
i
,
j
=
1
,
2
,...,
L
−
1
,
2
−
(
L
−
1
)
4
1
2
(
±
e
1
±
e
2
···±
e
L
−
1
±
e
L
)
L
=
6
,
7
,
8
Let us look at each of these definitions a bit closer and see how they can be used to generate
lattices.
2, the four roots of the
D
2
algebra are
e
1
+
e
2
,
D
L
Let us start with the
D
L
lattice. For
L
=
e
1
−
e
2
,
e
1
+
e
2
, and
e
1
−
e
2
, or (1, 1),
. We can pick
any two independent vectors from among these four to form the basis set for the
D
2
lattice.
Suppose we picked (1, 1) and
−
−
(
1
,
−
1
), (
−
1
,
1
)
, and
(
−
1
,
−
1
)
. Then any integral combination of these vectors is a
lattice point. The resulting lattice is shown in Figure 10.24 in Chapter 10. Notice that the
sums of the coordinates are all even numbers. This makes finding the closest lattice point to
an input a relatively simple exercise.
(
1
,
−
1
)
1-dimensional vectors. However, if we
select any
L
independent vectors from this set, we will find that the points that are generated
all lie in an
L
-dimensional slice of the
L
A
L
The roots of the
A
L
lattices are described using
L
+
+
1-dimensional space. This can be seen from
Figure C.1
We can obtain an
L
-dimensional basis set from this using a simple algorithm descri
be
d
in [
151
]. In two dimensions, this results in the generation of the vectors (1,0) and
√
3
2
1
.
The resulting lattice is shown in Figure 10.25 in Chapter 10. To find the closest point to the
A
L
lattice, we use the fact that in the embedding of the lattice in
L
(
−
2
,
)
+
1 dimensions, the sum
