The Optimal Isodual Lattice Quantizer in Three Dimensions

0.07 MB·English

by  J. H. Conway

The Optimal Isodual Lattice Quantizer in Three Dimensions
J. H. Conway
Mathematics Department
Princeton University
Princeton, NJ 08544
7
N. J. A. Sloane
0
0 AT&T Shannon Labs
2
180 Park Avenue

n Florham Park, NJ 07932-0971
a
J
Email: [e-mail], [e-mail]

3

] Jan 02 2006.
T
N
Abstract
.
h
t
a
The mean-centered cuboidal (or m.c.c.) lattice is known to be the optimal packing and
m
covering among all isodual three-dimensional lattices. In this note we show that it is also the best
[
quantizer. It thus joins the isodual lattices Z, A and (presumably) D , E and the Leech lattice
2 4 8
1
in being simultaneously optimal with respect to all three criteria.
v
0
8 Keywords: quantizing, self-dual lattice, isodual lattice, f.c.c. lattice, b.c.c. lattice, m.c.c.
0
lattice
1
0
7 AMS 2000 Classification: 52C07 (11H55, 94A29)
0
/
h 1. Introduction
t
a
m
: An isodual lattice [10] is one that is geometrically similar to its dual. Let Λ be an n-dimensional
v
i lattice andlet V denotethe Voronoi cell containing the origin. We may assumeΛ is scaled sothat
X
r V has unit volume. Three important parameters of Λ are the packing radius (the in-radius of V),
a
the covering radius (the circum-radius of V) and its quantization error, which is the normalized
second moment
1
G := x x dx (1)
n Z ·
V
(see [8], also [11], [12], [13]).
It is known that the face-centered cubic (or f.c.c.) lattice A has the largest packing radius of
3
any three-dimensional lattice (Gauss), while its dual, the body-centered cubic (or b.c.c.) lattice
A has both the smallest covering radius [1] and the smallest quantization error [2]. In [10] it was
∗3
shown that among all isodual three-dimensional lattices, the mean-centered cuboidal (or m.c.c.)
lattice has the largest packing radius and the smallest covering radius.
The m.c.c. lattice, denoted here by M , has Gram matrix
3
1+√2 1 1
− −
1  
1 1+√2 1 √2 ,
2 − −
 
 1 1 √2 1+√2 
 
− −
determinant 1, packing radius 1 1 + 1 , center density δ = 0.1657... (which is between the
2q2 √2
values for the f.c.c. and b.c.c. lattices), covering radius 30.52 1.25 (again between the values for
−
the f.c.c. and b.c.c. lattices), kissing number 8 and automorphism group of order 16. The m.c.c.
lattice received a brief mention in [3] and was studied in more detail in [10]. It also arises from
the period matrix of the hyperelliptic Riemann surface w2 = z8 1 [4].
−
The purpose of this note is to prove:
Theorem 1. The m.c.c. lattice M is the optimal quantizer among all isodual three-dimensional
3
lattices.
In higher dimensions, less is known. The lattice D is optimal among all four-dimensional lat-
4
tice packings, andhas alower quantization errorthanany other four-dimensionallattice presently
known (see [8] for references). It is not the best four-dimensional lattice covering (A is better),
∗4
but it is isodual and may be optimal among isodual lattices with respect to all three criteria.
Similar remarks apply to the isodual eight-dimensional lattice E (again A is a better covering
8 ∗8
but is not isodual). In 24 dimensions the isodual Leech lattice is known to be the best lattice
packing [6], and may well also be the optimal covering and quantizer.
2. Proof of Theorem 1
We will specify three-dimensional lattices by giving Gram matrices and conorms (or Selling pa-
rameters) — cf. [7], [9], [10]. Itwas shown in [10] that, upto equivalence, indecomposable isodual
three-dimensional lattices of determinant 1 have Gram matrices of the form
2α αβ α(2 β)
β − − −
1  
αβ 2β 2β(1−α) , (2)
2 αβ  − α − α 
−  
 −α(2−β) −2β(α1−α) α2β+2α+α2β−4αβ 
2
where α, β are any real numbers satisfying 0 < α < 1, 0 < β < 1; and decomposable lattices have
Gram matrices
1 0 0
 0 α h  , (3)
−
0 h β
 − 
where α, β, h are any real numbers satisfying 0 2h α β, αβ h2 = 1.
≤ ≤ ≤ −
In the indecomposable case the nonzero conorms are:
α(2 β) αβ 2α(1 β)
p = − , p = , p = − ,
01 02 03
γ γ βγ
2β(1 α) 2(1 α)(1 β) β(2 α)
p = − , p = − − , p = − , (4)
12 13 23
αγ γ γ
where γ = 2 αβ; in the decomposable case they are:
−
p = 1, p = α h, p = β h, p = 0, p = 0, p = h. (5)
01 02 03 12 13 23
− −
The m.c.c. lattice corresponds to the case α= β = 2 √2 of Eqs. (2) and (4).
−
From [2] we know that the normalized second moment (1) for a decomposable or indecompos-
able three-dimensional lattice Λ is given by
DS +2S +K
1 2
G = , (6)
36D4/3
where D = det Λ,
S = p +p +p +p +p +p ,
1 01 02 03 12 13 23
S = p p p p +p p p p +p p p p ,
2 01 02 13 23 01 03 12 23 02 03 12 13
K = p p p (p +p +p )+p p p (p +p +p )
01 02 03 12 13 23 01 12 13 02 03 23
+p p p (p +p +p )+p p p (p +p +p ).
02 12 23 01 03 13 03 13 23 01 02 12
For our lattices, D = 1.
We first consider the indecomposable case. From (4), (6) we find that
f(α,β)
G = (7)
36αβ(2 αβ)4
−
where
f(α,β) = 3α5β5 8α4β4(α+β)+4α3β3(α2+10αβ +β2)
−
48α3β3(α+β)+8α2β2(3α2 +4αβ +3β2)+32α2β2(α+β)
−
8αβ(5α2 +6αβ +5β2)+16(α2 +αβ +β2).
−
3
It turns out that there is exactly one choice for (α,β) in the range 0 < α < 1, 0 < β < 1 for
which both partial derivatives ∂G and ∂G vanish, namely α = β = 2 √2, corresponding to the
∂α ∂β −
m.c.c. lattice. At all other points in the interior of this region, one of the two partial derivatives
does not vanish and so the point cannot be a local minimum. To show this we used the computer
algebra system Maple [5] to compute the numerators of ∂G and ∂G, giving a pair of simultaneous
∂α ∂β
equations in α and β, symmetrical in α and β. By eliminating β, we obtain a single equation for
α:
α(α 1)(α 2)(α2 +2α 2)(α2 4α+2)(α4 4α3+6α2 4)
− − − − − − ×
(57α6 220α5 102α4 +1448α3 1860α2 +832α 152) = 0.
× − − − −
Theonlyrootsintherange0< α < 1are2 √2,√3 1andtheroot0.9894... ofthesixth-degree
− −
factor. By symmetry, β must also take one of these three values. At only one of these nine points
do both partial derivatives vanish, namely α = β = 2 √2. At that point the matrix of second
−
partial derivatives is positive definite, showing that m.c.c. is a local minimum. The resulting
value of G is
17+4√2
= 0.0786696... ,
288
between the values for the b.c.c. and f.c.c. lattices, which are respectively
19 21/3
22/3 = 0.0785432... and = 0.0787450... .
384 16
Ontheboundaryoftheregionthelatticesareeitherdegenerate(ifαorβ is0)ordecomposable
(if α or β is 1). In the latter case we assume α β and find that there is a unique point where
≤
∂G and ∂G vanish, when α = √3 1, β = 1. This is the lattice Z 3 1/4A , for which
∂α ∂β − ⊕ − 2
G = 5√3 + 1 = 0.0812361.... It is not a local minimum.
162 36
It remains to consider the decomposable case. From (5), (6), we find that
1
G = αβ(α+β)+2(αβ 1)3/2 +2α+2β +1 . (8)
36 n − o
Again we solve ∂G = ∂G = 0, and find that the only possibilities in the range 0 α β are
∂α ∂β ≤ ≤
α = β = 1, G = 1 ; α = 1, β = 2, G = 1 ; and α = β = √3 1 (the decomposable lattice
12 12 −
mentioned above). None of these are local minima. This completes the proof. The proof also
shows that the m.c.c. lattice is the only isodual lattice where G has a local minimum.
4
References
[1] R. P. Bambah, On lattice coverings by spheres, Proc. Nat. Inst. Sci. India, 20 (1954), 25–52.
[2] E. S. Barnes and N. J. A. Sloane, The optimal lattice quantizer in three dimensions, SIAM
J. Discrete Appl. Math., series 34 (1983), 30–41.
[3] A.-M. Berg´e and J. Martinet, Sur un probl`eme de dualit´e li´e aux sph`eres en g´eom´etrie des
nombres, J. Number Theory, 32 (1989), 14–42.
[4] M. Bernstein and N. J. A. Sloane, Some lattices obtained from Riemann surfaces, in “Ex-
tremalRiemannSurfaces”(Contemporary Math.Vol. 201), J.R.QuineandP.Sarnak,Eds.,
Amer. Math. Soc., Providence, RI, 1997, pp. 29–32.
[5] B. W. Char et al., “Maple 9,” MapleSoft, Waterloo, Canada, 2003.
[6] H. Cohn and A. Kumar, The densest lattice in twenty-four dimensions, Electron. Res. An-
nounc. Amer. Math. Soc., 10 (2004), 58–67.
[7] J. H. Conway, assisted by P. Y. C. Fung, “The Sensual (Quadratic) Form,” Math. Assoc.
Amer., Washington, DC, 1997.
[8] J. H. Conway and N. J. A. Sloane, “Sphere Packings, Lattices and Groups,” Springer, NY,
1988; 3rd. ed., 1998.
[9] J. H. Conway and N. J. A. Sloane, Low-dimensional lattices VI: Voronoi reduction of three-
dimensional lattices, Proc. Royal Soc. London, A 436 (1991), 55–68.
[10] J. H. Conway and N. J. A. Sloane, On lattices equivalent to their duals, J. Number Theory,
48 (1994), 373–382.
[11] Q.DuandD.Wang, The optimal centroidal Voronoi tessellations and the Gersho’s conjecture
in the three-dimensional space, Comput. Math. Appl., 49 (2005), 355–1373.
[12] R. M. Gray and D. L. Neuhoff, Quantization, IEEE Trans. Inform. Theory, 44 (1998), 2325–
2383.
[13] D. S. Kim and M. R. Bell, Upper bounds on empirically optimal quantizers, IEEE Trans.
Inform. Theory, 49 (2003), 1037–1046.
5