On a spherical code in the space of spherical harmonics
We propose a new method for the construction of new “nice” configurations of vectors on the unit sphere $S^d$ with the use of spaces of spherical harmonics.
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Institute of Mathematics, NAS of Ukraine
2010
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508915452608512 |
|---|---|
| author | Bondarenko, A. V. Бондаренко, А. В. |
| author_facet | Bondarenko, A. V. Бондаренко, А. В. |
| author_sort | Bondarenko, A. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:40:12Z |
| description | We propose a new method for the construction of new “nice” configurations of vectors on the unit sphere $S^d$ with the use of spaces of spherical harmonics. |
| first_indexed | 2026-03-24T02:32:48Z |
| format | Article |
| fulltext |
UDC 517.5
A. V. Bondarenko (Nat. Taras Shevchenko Univ. Kyiv, Ukraine)
ON A SPHERICAL CODE
IN THE SPACE OF SPHERICAL HARMONICS
PRO SFERYÇNYJ KOD
U PROSTORI SFERYÇNYX HARMONIK
We propose a new method for the construction of new “nice” configurations of vectors on the unit
sphere S
d with the use of spaces of spherical harmonics.
Zaproponovano novyj metod dlq pobudovy novyx „harnyx” konfihuracij vektoriv na odynyçnij
sferi S
d
z vykorystannqm prostoriv sferyçnyx harmonik.
1. Introduction. This paper is inspired by classical book J. H. Conway and N. J. A.
Sloane [1] and recent paper of H. Cohn and A. Kumar [2]. The exceptional
arrangement of points on the spheres are discussed there. Especially interesting are
constructions coming from well known E8 lattice and Leech lattice Λ24 . The main
idea of the paper is to use these arrangements for construction new good arrangements
in the spaces of spherical harmonics Hk
d . Recently we have use dramatically the
calculations in these spaces to obtain new asymptotic existence bounds for spherical
designs, see [3]. Below we need a few facts on spherical harmonics. Let ∆ be the
Laplace operator in Rd +1
∆ =
∂
∂=
+
∑
2
2
1
1
x jj
d
.
We say that a polynomial P in Rd +1 is harmonic if ∆ P = 0. For integer k ≥ 1 the
restriction to Sd of a homogeneous harmonic polynomial of degree k is called a
spherical harmonic of degree k . The vector space of all spherical harmonics of degree
k will be denoted by Hk
d (see [4] for details). The dimension of Hk
d is given by
dim Hk
d =
2 1
1
1k d
k d
d k
k
+ −
+ −
+ −
.
Consider usual inner product in Hk
d
〈 〉P Q, : = P x Q x d xd
Sd
( ) ( ) ( )µ∫ ,
where µd x( ) is normalized Lebesgue measure on the unit sphere Sd . Now, for each
point x Sd∈ there exists a unique polynomial Px k
d∈H such that
〈 〉P Qx , = Q x( ) for all Q k
d∈H .
It is well known that P yx ( ) = g x y(( , )) , where g is a corresponding Gegenbauer po-
lynomial. Let Gx be normalized polynomial Px , that is Gx = P gx / ( ) /1 1 2 . Note
© A. V. BONDARENKO, 2010
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6 857
858 A. V. BONDARENKO
that 〈 〉G Gx x1 2
, = g x x g(( , )) ( )/1 2 1 . So, if we have some arrangement X =
= { }, ,x xN1 … on Sd with known distribution of inner products ( , )x xi j , then, for
each k, we have corresponding set GX = { }, ,G Gx xN1
… in Hk
d , also with known
distribution of inner products. Using this construction we will obtain in the next secti-
on the optimal antipodal spherical ( 35, 240, 1 / 7 ) code from minimal vectors of E 8
lattice. Here is the definition.
Definition 1. An antipodal set X = { }, ,x xN1 … on Sd is called antipodal
spherical ( , , )d N a+ 1 code, if ( , )x xi j ≤ a, for some a > 0 and for all xi ,
x Xj ∈ , i ≠ j, which are not antipodal. Such code is called optimal if for any
antipodal set Y = { }, ,y yN1 … on Sd there exists yi , y Yj ∈ , i ≠ j , which
are not antipodal and ( , )y yi j ≥ a .
In the other words, antipodal spherical ( , , )d N a+ 1 code is optimal if a is a mi-
nimal possible number for fixed N, d.
Definition 2. An antipodal set X = { }, ,x xN1 … on Sd forms spherical 3-
design if and only if
1
2
2
1N
x xi j
i j
N
( , )
, =
∑ =
1
1d +
.
Note, that for all x x SN
d
1, ,… ∈ the following inequality hold
1
2
2
1N
x xi j
i j
N
( , )
, =
∑ ≥
1
1d +
.
Another equivalent definition is the following:
The set of points x x SN
d
1, ,… ∈ is called a spherical 3-design if
P x d xd
Sd
( ) ( )µ∫ =
1
1N
P xi
i
N
( )
=
∑
for all algebraic polynomials in d + 1 variables and of total degree at most 3, where
µd is normalized Lebesgue measure on Sd .
Thus we will prove the following theorem.
Theorem 1. There exists an optimal antipodal spherical ( 35, 240, 1 / 7 ) code,
those vectors form spherical 3-design.
2. Construction and the proof of optimality. Proof of Theorem 1. Let X =
= { }, ,x x1 120… be any subset of 240 normalized minimal vectors of E8 lattice, such
that no pair of antipodal vectors presents in X . Take in the space H2
7 the polyno-
mials
G yxi
( ) = g x yi2(( , )) , i = 1 120, ,… ,
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6
ON A SPHERICAL CODE IN THE SPACE OF SPHERICAL HARMONICS 859
where g t2( ) =
8
7
1
7
2t − is a corresponding normalized Gegenbauer polynomial.
Since ( , )x xi j = 0 or ± 1 / 2 , for i ≠ j, then 〈 〉G Gx xi j
, = g x yi j2(( , )) = ± 1 / 7 ! It
looks really like a mystery the fact that g x xi j2(( , )) = const, for any different xi ,
x Xj ∈ . But exactly this is essential for the proof of optimality of our code. Since,
dim H2
7 = 35, then the points G Gx x1 120
, , ,… − … −G Gx x1 120
, , provide antipodal
spherical ( 35, 240, 1 / 7 ) code. Here is a proof of optimality. Take arbitrary antipodal
set of points Y = { }, ,y y1 240… in R35 . Then, the inequality
1
2402
2
1
240
( , )
,
y yi j
i j =
∑ ≥ 1 / 35,
implies that ( , )y yi j
2 ≥ 1 / 49, for some yi , y Yj ∈ , i ≠ j, which are not antipodal.
This immediately gives us an optimality of our construction. The other reason why it
works, that is our set is also spherical 3-design in R35 . We are still not able genera-
lize this construction even for Leech lattice Λ24 . We also don’t know whether the
construction described above is an optimal spherical ( 35, 240, 1 / 7 ) code.
Acknowledgement. The author would like to thank Professor Henry Cohn for the
fruitfull discussions on the paper.
1. Conway J. H., Sloane N. J. A. Sphere packings, lattices and groups. – New York: Springer, 1999. –
703 p.
2. Cohn H., Kumar A. Universally optimal distribution of points on spheres // J. Amer. Math. Soc. –
2007. – 20. – P. 99 – 148.
3. Bondarenko A., Viazovska M. Spherical designs via Brouwer fixed point theorem // SIAM J.
Discrete Math. – 2010. – 24. – P. 207 – 217.
4. Mhaskar H. N., Narcowich F. J., Ward J. D. Spherical Marcinkiewich – Zygmund inequalities
and positive quadrature // Math. Comp. – 2000. – 70. – P. 1113 – 1130.
Received 22.07.09
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 6
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| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:32:48Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/d0/5ea0db543553d7155cdb13f92b3b47d0.pdf |
| spelling | umjimathkievua-article-29172020-03-18T19:40:12Z On a spherical code in the space of spherical harmonics Про сферичний код у просторі сферичних гармонік Bondarenko, A. V. Бондаренко, А. В. We propose a new method for the construction of new “nice” configurations of vectors on the unit sphere $S^d$ with the use of spaces of spherical harmonics. Запропоновано повий метод для побудови нових „гарних" конфігурацій векторів на одиничній сфері $S^d$ з використанням просторів сферичних гармонік. Institute of Mathematics, NAS of Ukraine 2010-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2917 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 6 (2010); 857 – 859 Український математичний журнал; Том 62 № 6 (2010); 857 – 859 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2917/2585 https://umj.imath.kiev.ua/index.php/umj/article/view/2917/2586 Copyright (c) 2010 Bondarenko A. V. |
| spellingShingle | Bondarenko, A. V. Бондаренко, А. В. On a spherical code in the space of spherical harmonics |
| title | On a spherical code in the space of spherical harmonics |
| title_alt | Про сферичний код у просторі сферичних гармонік |
| title_full | On a spherical code in the space of spherical harmonics |
| title_fullStr | On a spherical code in the space of spherical harmonics |
| title_full_unstemmed | On a spherical code in the space of spherical harmonics |
| title_short | On a spherical code in the space of spherical harmonics |
| title_sort | on a spherical code in the space of spherical harmonics |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2917 |
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