A construction of spherical 3-designs
UDC 512.5 We give a construction for spherical 3-designs. This construction is a generalization of Bondarenko’s results.
Gespeichert in:
| Datum: | 2022 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2022
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/986 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507239827111936 |
|---|---|
| author | Miezaki, T. Miezaki, T. |
| author_facet | Miezaki, T. Miezaki, T. |
| author_sort | Miezaki, T. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2022-03-27T15:39:11Z |
| description | UDC 512.5
We give a construction for spherical 3-designs. This construction is a generalization of Bondarenko’s results. |
| doi_str_mv | 10.37863/umzh.v74i1.986 |
| first_indexed | 2026-03-24T02:06:09Z |
| format | Article |
| fulltext |
К О Р О Т К I П О В I Д О М Л Е Н Н Я
DOI: 10.37863/umzh.v74i1.986
UDC 512.5
T. Miezaki (Waseda Univ., Tokyo, Japan)
A CONSTRUCTION OF SPHERICAL \bfthree -DESIGNS*
ПОБУДОВА СФЕРИЧНИХ \bfthree -КОНСТРУКЦIЙ
We give a construction for spherical 3-designs. This construction is a generalization of Bondarenko’s results.
Наведено метод побудови сферичних 3-конструкцiй. Цей метод є узагальненням результатiв Бондаренка.
1. Introduction. This paper is inspired by [1], which gives an optimal antipodal spherical (35, 240,
1/7) code whose vectors form a spherical 3-design. To explain our results, we review the concept
of spherical t-designs and [1].
First, we explain the concept of spherical t-designs.
Definition 1.1 [3]. For a positive integer t, a finite nonempty set X in the unit sphere
Sd =
\bigl\{
x = (x1, . . . , xd+1) \in \BbbR d+1 | x21 + . . .+ x2d+1 = 1
\bigr\}
is called a spherical t-design in Sd if the following condition is satisfied:
1
| X|
\sum
x\in X
f(x) =
1
| Sd|
\int
Sd
f(x)d\sigma (x),
for all polynomials f(x) = f(x1, . . . , xd+1) of degree not exceeding t. Here, the right-hand side
involves the surface integral over the sphere and | Sd| , the volume of sphere Sd.
The meaning of spherical t-designs is that the average value of the integral of any polynomial of
degree up to t on the sphere can be replaced by its average value over a finite set on the sphere.
The following is an equivalent condition of the antipodal spherical designs.
Proposition 1.1 [6]. An antipodal set X = \{ x1, . . . , xN\} in Sd forms a spherical 3-design if
and only if
1
| X| 2
\sum
xi,xj\in X
(xi, xj)
2 =
1
d+ 1
.
An antipodal set X = \{ x1, . . . , xN\} in Sd forms a spherical 5-design if and only if
1
| X| 2
\sum
xi,xj\in X
(xi, xj)
2 =
1
d+ 1
,
1
| X| 2
\sum
xi,xj\in X
(xi, xj)
4 =
3
(d+ 3)(d+ 1)
.
(1.1)
Next, we review [1]. Let
\Delta =
d+1\sum
j=1
\partial 2
\partial x2j
.
* This paper was supported by JSPS KAKENHI (18K03217).
c\bigcirc T. MIEZAKI, 2022
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1 141
142 T. MIEZAKI
We say that a polynomial P in \BbbR d+1 is harmonic if \Delta P = 0. For integer k \geq 1, the restriction of
a homogeneous harmonic polynomial of degree k to Sd is called a spherical harmonic of degree k.
We denote by \mathrm{H}\mathrm{a}\mathrm{r}\mathrm{m}k(S
d) the vector space of the spherical harmonics of degree k. Note that (see,
for example, [6])
\mathrm{d}\mathrm{i}\mathrm{m}\mathrm{H}\mathrm{a}\mathrm{r}\mathrm{m}k(S
d) =
2k + d - 1
k + d - 1
\biggl(
d+ k - 1
k
\biggr)
.
For P,Q \in \mathrm{H}\mathrm{a}\mathrm{r}\mathrm{m}k(S
d), we denote by \langle P,Q\rangle the usual inner product
\langle P,Q\rangle :=
\int
Sd
P (x)Q(x)d\sigma (x),
where d\sigma (x) is a normalized Lebesgue measure on the unit sphere Sd. For x \in Sd, there exists
Px \in \mathrm{H}\mathrm{a}\mathrm{r}\mathrm{m}k(S
d) such that
\langle Px, Q\rangle = Q(x) for all Q \in \mathrm{H}\mathrm{a}\mathrm{r}\mathrm{m}k(S
d).
It is known that
Px(y) = gk,d((x, y)),
where gk,d is a Gegenbauer polynomial. Let
Gx =
Px
gk,d(1)1/2
.
We remark that
\langle Gx, Gy\rangle =
gk,d((x, y))
gk,d(1)
.
(For a detailed explanation of Gegenbauer polynomials, see [6].) Therefore, if we have a set X =
= \{ x1, . . . , xN\} in Sd, then we obtain the set GX = \{ Gx1 , . . . , GxN \} in SdimHarmk(S
d) - 1.
Let X = \{ x1, . . . , x120\} be an arbitrary subset of 240 normalized minimum vectors of the E8
lattice such that no pair of antipodal vectors is present in X. Set Px(y) = g2,7((x, y)). A. V. Bon-
darenko in [1] showed that GX \cup - GX is an optimal antipodal spherical (35, 240, 1/7) code whose
vectors form a spherical 3-design, where
- GX := \{ - Gx | Gx \in GX\} .
Furthermore, A. V. Bondarenko in [1] showed that GX \cup - GX is a spherical 3-design, using the
special properties of the E8 lattice. However, this fact is an example that extends to a more general
setting as follows. The spherical 3-design obtained by A. V. Bondarenko in [1] is a special case of
our main result, which is presented as the following theorem.
Theorem 1.1. Let X be a finite subset of sphere Sd satisfying the condition (1.1). We set
Px(y) = g2,d((x, y)). Then GX \cup - GX is a spherical 3-design in SdimHarm2(Sd) - 1.
We denote by \widetilde GX the set GX \cup - GX defined in Theorem 1.1.
Corollary 1.1. 1. Let X be a spherical 4-design in Sd. Then \widetilde GX is a spherical 3-design
in SdimHarm2(Sd) - 1.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
A CONSTRUCTION OF SPHERICAL 3-DESIGNS 143
2. Let X be a spherical 4-design in Sd and an antipodal set. Let X \prime be an arbitrary subset
of X with | X \prime | = | X| /2 such that no pair of antipodal vectors is present in X \prime . Then \widetilde GX\prime is a
spherical 3-design in SdimHarm2(Sd) - 1.
In Section 2, we give a proof of Theorem 1.1. In Section 3, we give some examples.
2. Proof of Theorem 1.1. Let X = \{ x1, . . . , xN\} be in Sd and GX = \{ Gx1 , . . . , GxN \} be in
\mathrm{H}\mathrm{a}\mathrm{r}\mathrm{m}2(S
d). By Proposition 1.1, we have
1
| X| 2
\sum
xi,xj\in X
(xi, xj)
2 =
1
d+ 1
,
1
| X| 2
\sum
xi,xj\in X
(xi, xj)
4 =
3
(d+ 3)(d+ 1)
,
since X is a spherical 4-design. We have the following Gegenbauer polynomial of degree 2 on Sd :
g2,d(x) =
d+ 1
d
x2 - 1
d
.
It is enough to show that
1
| X| 2
\sum
xi,xj\in X
\langle Gxi , Gxj \rangle 2 =
2
d(d+ 3)
,
since
\mathrm{d}\mathrm{i}\mathrm{m}\mathrm{H}\mathrm{a}\mathrm{r}\mathrm{m}2(S
d) =
d+ 3
d+ 1
\biggl(
d+ 1
2
\biggr)
=
d(d+ 3)
2
and GX \cup - GX is an antipodal set. We remark that if X is a spherical t-design, then X \cup - X is
also a spherical t-design.
In fact,
1
| X| 2
\sum
xi,xj\in X
\langle Gxi , Gxj \rangle 2 =
1
| X| 2
\sum
xi,xj\in X
g2,d((xi, xj))
2 =
=
1
| X| 2
\sum
xi,xj\in X
\biggl(
(d+ 1)2
d2
(xi, xj)
4 - 2
d+ 1
d2
(xi, xj)
2 +
1
d2
\biggr)
=
=
(d+ 1)2
d2
3
(d+ 3)(d+ 1)
- 2
d+ 1
d2
1
d+ 1
+
1
d2
=
2
d(d+ 3)
.
Therefore, if X = \{ x1, . . . , xN\} is a spherical 4-design, then GX \cup - GX is a spherical 3-design.
Theorem 1.1 is proved.
3. Examples. In this section, we give some examples of using Theorem 1.1.
First we recall the concept of a strongly perfect and spherical (d+ 1, N, a) code.
Definition 3.1 [6]. A lattice L is called strongly perfect if the minimum vectors of L form a
spherical 5-design.
Definition 3.2 [2]. An antipodal set X = \{ x1, . . . , xN\} in Sd is called an antipodal spherical
(d+ 1, N, a) code if | (xi, xj)| \leq a for some a > 0 and all xi, xj \in X, i \not = j, are not antipodal.
Next we give some examples.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
144 T. MIEZAKI
Example 3.1. The strongly perfect lattices whose ranks are less than 12 have been classified
[4, 5]. Such lattices whose ranks are greater than 1 are as follows:
A2, D4, E6, E
\sharp
6, E7, E
\sharp
7, E8, K10, K
\sharp
10, CT12.
(For a detailed explanation, see [4, 5].) Let L be one of the above lattices and X be the minimum
vectors of L. Then, let X \prime be an arbitrary subset of X with | X \prime | = | X| /2 such that no pair of
antipodal vectors is present in X \prime .
By Corollary 1.1, GX \cup - GX is a spherical 3-design in Sd, where d is as follows:
L (d+ 1, N, a) code | (xi, xj)| | \langle Gxi , Gxj \rangle |
A2 (2, 6, 1/2) \{ 1/2\} \{ 1/2\}
D4 (9, 24, 1/3) \{ 0, 1/2\} \{ 0, 1/3\}
E6 (20, 72, 1/5) \{ 0, 1/2\} \{ 1/10, 1/5\}
E\sharp
6 (20, 54, 1/8) \{ 1/4, 1/2\} \{ 1/10, 1/8\}
E7 (27, 126, 1/6) \{ 0, 1/2\} \{ 1/8, 1/6\}
E\sharp
7 (27, 56, 1/27) \{ 1/3\} \{ 1/27\}
E8 (35, 240, 1/7) [1] \{ 0, 1/2\} \{ 1/7\}
K10 (54, 276, 1/6) \{ 0, 1/4, 1/2\} \{ 1/24, 1/9, 1/6\}
K\sharp
10 (54, 54, 1/6) \{ 1/8, 1/4, 1/2\} \{ 1/24, 3/32, 1/6\}
CT12 (77, 756, 2/11) \{ 0, 1/4, 1/2\} \{ 1/44, 1/11, 2/11\}
Example 3.2. Let X be the minimum vectors of the Barnes – Wall lattice of rank 16, and let X \prime
be an arbitrary subset of X with | X \prime | = | X| /2 such that no pair of antipodal vectors is present in
X \prime . We remark that X is a spherical 7-design.
By Corollary 1.1, GX \cup - GX is a spherical 3-design in Sd, where d is as follows:
L (d+ 1, N, a) code | (xi, xj)| | \langle Gxi , Gxj \rangle |
\mathrm{B}\mathrm{W}16 (135, 4320, 1/5) \{ 0, 1/4, 1/2\} \{ 0, 1/15, 1/5\}
Example 3.3. Let X be the minimum vectors of the Leech lattice, and let X \prime be an arbitrary
subset of X with | X \prime | = | X| /2 such that no pair of antipodal vectors is present in X \prime . We remark
that X is a spherical 11-design.
By Corollary 1.1, GX \cup - GX is a spherical 3-design in Sd, where d is as follows:
L (d+ 1, N, a) code | (xi, xj)| | \langle Gxi , Gxj \rangle |
\mathrm{L}\mathrm{e}\mathrm{e}\mathrm{c}\mathrm{h} (299, 196560, 5/23) \{ 0, 1/4, 1/2\} \{ 1/46, 1/23, 5/23\}
References
1. A. V. Bondarenko, On a spherical code in the space of spherical harmonics, Ukr. Math. J., 62, № 6, 993 – 996 (2010).
2. J. H. Conway, N. J. A. Sloane, Sphere packings lattices and groups, third ed., Springer, New York (1999).
3. P. Delsarte, J.-M. Goethals, J. J. Seidel, Spherical codes and designs, Geom. Dedicata, 6, 363 – 388 (1977).
4. G. Nebe, B. Venkov, The strongly perfect lattices of dimension 10, Colloque International de Theorie des Nombres
(Talence, 1999), J. Theor. Nombres Bordeaux, 12, № 2, 503 – 518 (2000).
5. G. Nebe, B. Venkov, Low-dimensional strongly perfect lattices. I. The 12-dimensional case, Enseign. Math., 51,
№ 1-2, 129 – 163 (2005).
6. B. Venkov, Réseaux et designs sphériques. Réseaux euclidiens, designs sphériques et formes modulaires, Monogr.
Enseign. Math., 37, 10 – 86 (2001).
Received 25.06.19
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 1
|
| id | umjimathkievua-article-986 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:06:09Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/94/924eb576618f8f0db9d65ac5f818c794.pdf |
| spelling | umjimathkievua-article-9862022-03-27T15:39:11Z A construction of spherical 3-designs A construction of spherical 3-designs Miezaki, T. Miezaki, T. Spherical designs Lattices Spherical harmonics Spherical designs Lattices Spherical harmonics UDC 512.5 We give a construction for spherical 3-designs. This construction is a generalization of Bondarenko’s results. УДК 512.5 Побудова сферичних 3 -конструкцiй Наведено метод побудови сферичних $3$-конструкцій. Цей метод є узагальненням результатів Бондаренка. Institute of Mathematics, NAS of Ukraine 2022-01-24 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/986 10.37863/umzh.v74i1.986 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 1 (2022); 141 - 144 Український математичний журнал; Том 74 № 1 (2022); 141 - 144 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/986/9184 Copyright (c) 2021 Tsuyoshi Miezaki |
| spellingShingle | Miezaki, T. Miezaki, T. A construction of spherical 3-designs |
| title | A construction of spherical 3-designs |
| title_alt | A construction of spherical 3-designs |
| title_full | A construction of spherical 3-designs |
| title_fullStr | A construction of spherical 3-designs |
| title_full_unstemmed | A construction of spherical 3-designs |
| title_short | A construction of spherical 3-designs |
| title_sort | construction of spherical 3-designs |
| topic_facet | Spherical designs Lattices Spherical harmonics Spherical designs Lattices Spherical harmonics |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/986 |
| work_keys_str_mv | AT miezakit aconstructionofspherical3designs AT miezakit aconstructionofspherical3designs AT miezakit constructionofspherical3designs AT miezakit constructionofspherical3designs |