Universal optimality of $T$-avoiding spherical codes and designs

D. D. Cherkashin; P. D. Dragnev; P. G. Boyvalenkov

arxiv: 2501.13906 · v2 · pith:X7SID7CBnew · submitted 2025-01-23 · 🧮 math.CO · cs.IT· math.IT· math.MG

Universal optimality of T-avoiding spherical codes and designs

P. G. Boyvalenkov , D. D. Cherkashin , P. D. Dragnev This is my paper

Pith reviewed 2026-05-23 04:42 UTC · model grok-4.3

classification 🧮 math.CO cs.ITmath.ITmath.MG

keywords spherical codesT-avoiding codesuniversal optimalityLeech latticeBarnes-Wall latticestrongly regular graphsspherical designs

0 comments

The pith

Certain Leech and Barnes-Wall lattice codes are universally optimal among all T-avoiding spherical codes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines T-avoiding spherical codes as those whose inner products avoid a prescribed open set T inside [-1,1). It proves that particular codes taken from the minimal vectors of the Leech lattice, the Barnes-Wall lattice, and certain strongly regular graphs achieve universal optimality inside this restricted class. Universal optimality here means that the codes minimize a broad family of potential-energy functionals among all T-avoiding competitors of the same dimension. The same codes are shown to be tight T-avoiding designs and, in several cases, to attain the largest possible size for given dimension and minimum distance.

Core claim

We show that certain codes found in the minimal vectors of the Leech lattice, as well as the minimal vectors of the Barnes-Wall lattice and codes derived from strongly regular graphs, are universally optimal in the restricted class of T-avoiding codes. We also extend a result of Delsarte-Goethals-Seidel about codes with three inner products and derive that these codes are minimal (tight) T-avoiding spherical designs of fixed dimension and strength.

What carries the argument

The T-avoiding restriction: an open set T subset [-1,1) is chosen so the inner products realized by the lattice or graph codes lie entirely outside T, turning the usual universal-optimality question into a constrained one on the complement of T.

If this is right

The codes minimize energy for every completely monotonic potential whose support lies outside T.
They are minimal T-avoiding designs of the given strength and dimension.
In several cases they also realize the largest possible cardinality among T-avoiding codes with the given minimum distance.
The three-inner-product case recovers and generalizes the Delsarte-Goethals-Seidel theorem inside the (α,β)-avoiding subclass.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same avoidance technique might produce new upper bounds on the size of spherical codes when certain angles are forbidden by application constraints.
If the method extends to other root lattices or association schemes, it could classify additional families of optimal avoiding codes.
One could test whether the optimality persists when T is enlarged slightly while still excluding the code's inner products.

Load-bearing premise

The open set T can be chosen so that none of the inner products realized by the candidate codes fall inside T.

What would settle it

Exhibit a single T-avoiding spherical code of the same dimension whose energy is strictly smaller than that of the claimed optimal code for at least one admissible potential function.

read the original abstract

Given an open set $T\subset [-1,1)$, we introduce the concepts of $T$-avoiding spherical codes and designs, that is, spherical codes that have no inner products in the set $T$. We show that certain codes found in the minimal vectors of the Leech lattice, as well as the minimal vectors of the Barnes--Wall lattice and codes derived from strongly regular graphs, are universally optimal in the restricted class of $T$-avoiding codes. We also extend a result of Delsarte--Goethals--Seidel about codes with three inner products $\alpha, \beta, \gamma$ (in our terminology $(\alpha,\beta)$-avoiding $\gamma$-codes). Parallel to the notion of tight spherical designs, we also derive that these codes are minimal (tight) $T$-avoiding spherical designs of fixed dimension and strength. In some cases, we also find that codes under consideration have maximal cardinality in their $T$-avoiding class for given dimension and minimum distance.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

They define T-avoiding spherical codes and show that Leech, Barnes-Wall, and certain graph codes are universally optimal inside that restricted class.

read the letter

The main thing to know is that the paper introduces T-avoiding spherical codes (no inner products in a chosen open set T) and proves that several classical configurations remain universally optimal when the search is limited to that class. It also extends the Delsarte-Goethals-Seidel result on three-inner-product codes and shows the examples are tight T-avoiding designs, with some maximality statements for cardinality as well. The concrete objects are the minimal vectors of the Leech and Barnes-Wall lattices plus codes from strongly regular graphs. This is new work; the T-avoiding restriction has not been studied in this form before, and tying it directly to these well-known lattices and graphs gives the claims something solid to stand on rather than abstract generality. The extension of the classical DGS theorem looks like a clean rephrasing that fits the new language. The soft spot is that T must be chosen so the target codes avoid it, which is built into the setup. This makes the optimality statement hold but keeps the result inside a tailored subclass; it does not claim unrestricted optimality. That is not a flaw in the logic, just a limit on how far the conclusion travels. The abstract gives no explicit T values or proof outlines, so the technical details of the energy arguments and the tightness claims would need checking in the full text, but the overall framing does not show circularity or hidden fitting. This is for people already working in spherical codes, universal optimality, and combinatorial designs. A reader who follows lattice-based codes or Delsarte-type bounds would get direct value from the new class and the listed examples. I would send it to peer review because the idea is focused, the applications are to established objects, and the restricted-class claims are stated clearly enough to be checked.

Referee Report

2 major / 2 minor

Summary. The paper introduces T-avoiding spherical codes and designs for an open set T ⊂ [-1,1). It claims that certain codes realized by minimal vectors of the Leech lattice, minimal vectors of the Barnes-Wall lattice, and codes derived from strongly regular graphs are universally optimal among T-avoiding codes (for T chosen so the codes avoid T). It extends the Delsarte-Goethals-Seidel theorem on codes with three inner products to the T-avoiding setting, shows the codes are tight (minimal) T-avoiding spherical designs of fixed dimension and strength, and establishes maximal cardinality in the T-avoiding class for given dimension and minimum distance in some cases.

Significance. If the proofs hold, the work is significant for extending universal optimality results to well-defined restricted classes of spherical codes, supplying concrete examples tied to the Leech and Barnes-Wall lattices and to strongly regular graphs. The parallel development of tight T-avoiding designs and the explicit restriction to T-avoiding codes (rather than an overclaim of unrestricted optimality) are strengths; the results are falsifiable via explicit energy comparisons within the restricted class.

major comments (2)

[Theorems on lattice codes (around the statements following the definition of T-avoiding codes)] The central optimality claims are load-bearing on the existence of suitable open T that the codes avoid; the manuscript should state the explicit form of each such T (or the construction that produces it) in the statements of the main theorems on the Leech and Barnes-Wall examples so that avoidance can be directly verified.
[Section containing the DGS extension (likely §3 or §4)] The extension of the Delsarte-Goethals-Seidel result to (α,β)-avoiding γ-codes requires a precise statement of how the original linear-programming or association-scheme argument is adapted when T is an arbitrary open set containing none of the realized inner products; the current sketch leaves open whether additional constraints on T are needed for the bound to remain valid.

minor comments (2)

[Introduction] Notation for the open set T and the avoidance condition is introduced clearly in the abstract but should be repeated with a displayed definition early in the introduction for readers who begin with the main text.
[Abstract and concluding remarks] The abstract states that 'in some cases' the codes achieve maximal cardinality; the manuscript should indicate which of the three families (Leech, Barnes-Wall, graph-derived) satisfy this and under what additional hypotheses on dimension or distance.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the referee's careful and constructive review. We address each major comment below and will revise the manuscript to incorporate the suggested clarifications.

read point-by-point responses

Referee: [Theorems on lattice codes (around the statements following the definition of T-avoiding codes)] The central optimality claims are load-bearing on the existence of suitable open T that the codes avoid; the manuscript should state the explicit form of each such T (or the construction that produces it) in the statements of the main theorems on the Leech and Barnes-Wall examples so that avoidance can be directly verified.

Authors: We agree that explicit statements of T will strengthen the presentation. In the revised version, each main theorem on the Leech and Barnes-Wall examples will include the explicit construction of the corresponding open set T, defined as the complement in [-1,1) of the finite set of inner products realized by the code. This makes the T-avoiding property directly verifiable from the theorem statement without reference to later sections. revision: yes
Referee: [Section containing the DGS extension (likely §3 or §4)] The extension of the Delsarte-Goethals-Seidel result to (α,β)-avoiding γ-codes requires a precise statement of how the original linear-programming or association-scheme argument is adapted when T is an arbitrary open set containing none of the realized inner products; the current sketch leaves open whether additional constraints on T are needed for the bound to remain valid.

Authors: We will expand the sketch into a precise outline. The adaptation requires no constraints on T beyond openness and disjointness from the realized inner products: the linear-programming or association-scheme inequalities are applied exclusively to the allowed inner products, and the nonnegativity of the relevant polynomials or kernels is preserved under this restriction. The revised section will state this explicitly and supply the full adaptation steps. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces T-avoiding spherical codes (open set T subset [-1,1) with no inner products in T) and proves that specific codes from Leech lattice minimal vectors, Barnes-Wall lattice, and strongly regular graphs are universally optimal within the restricted T-avoiding class, for T chosen to exclude the codes' realized inner products. It also extends the external Delsarte-Goethals-Seidel result on three-inner-product codes and derives tightness/minimality properties for T-avoiding designs. These are statements of restricted optimality whose truth depends only on energy minimization inside the subclass; no equation reduces a claimed prediction to a fitted input by construction, no self-definitional loop appears (T is chosen externally to the codes' properties), and load-bearing steps cite external lattice facts and the DGS theorem rather than prior work by the same authors. The derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper works inside the established theory of spherical codes and designs; it introduces no new free parameters, no new postulated entities, and relies only on standard mathematical background.

axioms (1)

standard math Standard inner-product geometry on the unit sphere and the definition of spherical designs via moment conditions
Invoked throughout the theory of spherical codes and designs; the abstract assumes this background.

pith-pipeline@v0.9.0 · 5726 in / 1238 out tokens · 24967 ms · 2026-05-23T04:42:17.180161+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 1.4 and Theorem 3.4: T-avoiding universally optimal means Eh-minimizer among |C|=N codes with I(C)∩T=∅; LP uses f≤h on [−1,1]∖T and fi≥0 (i≥1).
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Alexander-duality style dimension forcing is absent; dimensions 16/22/23/48 are taken as given from lattices.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

[1]

Bannai, N

E. Bannai, N. J. A. Sloane, Uniqueness of certain spheric al codes, Canad. J. Math. 33 (1981) 437–449

work page 1981
[2]

Bilyk, A

D. Bilyk, A. Glazyrin, R. Matzke, J. Park, O. Vlasiuk, Ene rgy on spheres and discreteness of minimizing measures, J. Funct. Anal., 280 #11 (2021) 28 pp

work page 2021
[3]

Borodachov, P

S. Borodachov, P . Boyvalenkov, P . Dragnev, D. Hardin, E. Saff, M. Stoyanova, Optimal polarization pairs of spherical codes embedded in E8 and Leech lattices, submitted (2025)

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V ., Hardin, D

Borodachov, S. V ., Hardin, D. P ., Saff, E. B., Discrete En ergy on Rectiﬁable Sets, Springer Monographs in Math- ematics, Springer, 2019

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Boyvalenkov, Nonexistence of certain symmetric sphe rical codes, Des

P . Boyvalenkov, Nonexistence of certain symmetric sphe rical codes, Des. Codes Crypt. 3 (1993) 69–74

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[6]

Boyvalenkov, D

P . Boyvalenkov, D. Cherkashin, The kissing number in 48 d imensions for codes with certain forbidden distances is 52 416 000, Results in Mathematics, 80 (2025) art. 3

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Boyvalenkov, P

P . Boyvalenkov, P . Dragnev, D. Hardin, E. Saff, M. Stoyan ova, Universal lower bounds for potential energy of spherical codes, Constr . Approx.44 (2016) 385–415

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Boyvalenkov, P

P . Boyvalenkov, P . Dragnev, D. Hardin, E. Saff, M. Stoyan ova, Universal minima of discrete potentials for sharp spherical codes, Rev. Mat. Iberoam. (2024), published onli ne ﬁrst. T -A VOIDING SPHERICAL CODES AND DESIGNS 31

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Boyvalenkov, P

P . Boyvalenkov, P . Dragnev, Energy of codes with forbidden distances in 48 dimensions, accepted in Applied and Numerical Harmonic Analysis, a Springer book series

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J. S. Brauchart, Complete Minimal Logarithmic Energy A symptotics for Points in a Compact Interval: A Conse- quence of the Discriminant of Jacobi Polynomials, Constr . Approx.59, (2024) 717–735

work page 2024
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J. S. Brauchart, P . J. Grabner, Distributing many point s on spheres: minimal energy and designs, J. Complexity 31 (2015) 293–326

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H. Cohn, A. Kumar, Universally optimal distribution of points on spheres, J. Amer . Math. Soc., 20 (2006) 99–148

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H. Cohn, A. Kumar, Uniqueness of the (22, 891, 1/4) spherical code, New York J. Math. 13 (2007) 147–157,

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H. Cohn, A. Kumar, S. D. Miller, D. Radchenko, M. Viazovs ka, The sphere packing problem in dimension 24, Ann. of Math. 185 (2017) 1017–1033

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H. Cohn, A. Kumar, S. D. Miller, D. Radchenko, M. Viazovs ka, Universal optimality of the E8 and Leech lattices and interpolation formulas, Ann. of Math. 196 (2022) 983–1082

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H. Cohn, D. de Laat, N. Leijenhorst, Optimality of spher ical codes via exact semideﬁnite programming bounds, arXiv:2403.16874

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J. H. Conway, N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer V erlag, New Y ork, Third edition, 1999

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P . D. Dragnev, D. A. Legg, D. W. Townsend, Discrete logar ithmic energy on the sphere, Paciﬁc J. Math. 207 (2002) 345–357

work page 2002
[19]

Delsarte, J.-M

P . Delsarte, J.-M. Goethals, J. J. Seidel, Spherical codes and designs, Geom. Dedicata 6 (1977) 363–388

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[20]

Ericson, V

T. Ericson, V . Zinoviev, Codes on Euclidean spheres, El sevier, 2001

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Fazekas, V

G. Fazekas, V . I. Levenshtein, On upper bounds for code d istance and covering radius of designs in polynomial metric spaces, J. Comb. Theory A, 70 (1995) 267–288

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Gasper, Linearization of the product of Jacobi polyn omials I, Canad

G. Gasper, Linearization of the product of Jacobi polyn omials I, Canad. J. Math. 22 (1970) 171–175

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Gonc ¸alves, G

F. Gonc ¸alves, G. V edana, Sphere packings in Euclideanspace with forbidden distances, arXiv:2308.03925

work page arXiv
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Hardin, E

D. Hardin, E. Saff, Discretizing manifolds via minimum energy points, Notices of the American Math. Soc. , Nov. 2004, 1186–1194

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G. A. Kabatianskii, V . I. Levenshtein, Bounds for packi ngs on a sphere and in space, Probl. Inform. Transm. 14 (1978) 1–17

work page 1978
[26]

A. V . Kolushov, V . A. Y udin, Extremal dispositions of points on the sphere, Anal. Math. 23 (1997) 25–34

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V . I. Levenshtein, Universal bounds for codes and desig ns, in Handbook of Coding Theory, V . S. Pless and W. C. Huffman, Eds., Elsevier, Amsterdam, Ch. 6, 499–648 (1998)

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Nebe, A fourth extremal even unimodular lattice of di mension 48, Discr

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Niemeier, Deﬁnite quadratische Formen der Dimen sion 24 und Diskriminante I, J

H.-V . Niemeier, Deﬁnite quadratische Formen der Dimen sion 24 und Diskriminante I, J. Numb. Th. 5 (1973) 142–178

work page 1973
[30]

Saff, A.B.J

E.B. Saff, A.B.J. Kuijlaars, Distributing many points on a sphere, The Mathematical Intelligencer, 19 (1997) 5–11

work page 1997
[31]

R. E. Schwartz, Five Point Energy Minimization: A Synop sis, Constr. Approx. 51 (2020) 537–564

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[32]

I. J. Schoenberg, Positive deﬁnite functions on sphere s, Duke Math. J. 9 (1942) 96–108

work page 1942
[33]

Viazovska, The sphere packing problem in dimension 8 , Ann

M. Viazovska, The sphere packing problem in dimension 8 , Ann. of Math. 185 (2017) 991–1015. INSTITUTE OF MATHEMATICS AND INFORMATICS , B ULGARIAN ACADEMY OF SCIENCES , 8 G B ONCHEV STR ., 1113 S OFIA , B ULGARIA Email address: peter@math.bas.bg INSTITUTE OF MATHEMATICS AND INFORMATICS , B ULGARIAN ACADEMY OF SCIENCES , 8 G B ONCHEV STR ., 1113 S OFIA , B...

work page 2017

[1] [1]

Bannai, N

E. Bannai, N. J. A. Sloane, Uniqueness of certain spheric al codes, Canad. J. Math. 33 (1981) 437–449

work page 1981

[2] [2]

Bilyk, A

D. Bilyk, A. Glazyrin, R. Matzke, J. Park, O. Vlasiuk, Ene rgy on spheres and discreteness of minimizing measures, J. Funct. Anal., 280 #11 (2021) 28 pp

work page 2021

[3] [3]

Borodachov, P

S. Borodachov, P . Boyvalenkov, P . Dragnev, D. Hardin, E. Saff, M. Stoyanova, Optimal polarization pairs of spherical codes embedded in E8 and Leech lattices, submitted (2025)

work page 2025

[4] [4]

V ., Hardin, D

Borodachov, S. V ., Hardin, D. P ., Saff, E. B., Discrete En ergy on Rectiﬁable Sets, Springer Monographs in Math- ematics, Springer, 2019

work page 2019

[5] [5]

Boyvalenkov, Nonexistence of certain symmetric sphe rical codes, Des

P . Boyvalenkov, Nonexistence of certain symmetric sphe rical codes, Des. Codes Crypt. 3 (1993) 69–74

work page 1993

[6] [6]

Boyvalenkov, D

P . Boyvalenkov, D. Cherkashin, The kissing number in 48 d imensions for codes with certain forbidden distances is 52 416 000, Results in Mathematics, 80 (2025) art. 3

work page 2025

[7] [7]

Boyvalenkov, P

P . Boyvalenkov, P . Dragnev, D. Hardin, E. Saff, M. Stoyan ova, Universal lower bounds for potential energy of spherical codes, Constr . Approx.44 (2016) 385–415

work page 2016

[8] [8]

Boyvalenkov, P

P . Boyvalenkov, P . Dragnev, D. Hardin, E. Saff, M. Stoyan ova, Universal minima of discrete potentials for sharp spherical codes, Rev. Mat. Iberoam. (2024), published onli ne ﬁrst. T -A VOIDING SPHERICAL CODES AND DESIGNS 31

work page 2024

[9] [9]

Boyvalenkov, P

P . Boyvalenkov, P . Dragnev, Energy of codes with forbidden distances in 48 dimensions, accepted in Applied and Numerical Harmonic Analysis, a Springer book series

work page

[10] [10]

J. S. Brauchart, Complete Minimal Logarithmic Energy A symptotics for Points in a Compact Interval: A Conse- quence of the Discriminant of Jacobi Polynomials, Constr . Approx.59, (2024) 717–735

work page 2024

[11] [11]

J. S. Brauchart, P . J. Grabner, Distributing many point s on spheres: minimal energy and designs, J. Complexity 31 (2015) 293–326

work page 2015

[12] [12]

H. Cohn, A. Kumar, Universally optimal distribution of points on spheres, J. Amer . Math. Soc., 20 (2006) 99–148

work page 2006

[13] [13]

H. Cohn, A. Kumar, Uniqueness of the (22, 891, 1/4) spherical code, New York J. Math. 13 (2007) 147–157,

work page 2007

[14] [14]

H. Cohn, A. Kumar, S. D. Miller, D. Radchenko, M. Viazovs ka, The sphere packing problem in dimension 24, Ann. of Math. 185 (2017) 1017–1033

work page 2017

[15] [15]

H. Cohn, A. Kumar, S. D. Miller, D. Radchenko, M. Viazovs ka, Universal optimality of the E8 and Leech lattices and interpolation formulas, Ann. of Math. 196 (2022) 983–1082

work page 2022

[16] [16]

H. Cohn, D. de Laat, N. Leijenhorst, Optimality of spher ical codes via exact semideﬁnite programming bounds, arXiv:2403.16874

work page arXiv

[17] [17]

J. H. Conway, N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer V erlag, New Y ork, Third edition, 1999

work page 1999

[18] [18]

P . D. Dragnev, D. A. Legg, D. W. Townsend, Discrete logar ithmic energy on the sphere, Paciﬁc J. Math. 207 (2002) 345–357

work page 2002

[19] [19]

Delsarte, J.-M

P . Delsarte, J.-M. Goethals, J. J. Seidel, Spherical codes and designs, Geom. Dedicata 6 (1977) 363–388

work page 1977

[20] [20]

Ericson, V

T. Ericson, V . Zinoviev, Codes on Euclidean spheres, El sevier, 2001

work page 2001

[21] [21]

Fazekas, V

G. Fazekas, V . I. Levenshtein, On upper bounds for code d istance and covering radius of designs in polynomial metric spaces, J. Comb. Theory A, 70 (1995) 267–288

work page 1995

[22] [22]

Gasper, Linearization of the product of Jacobi polyn omials I, Canad

G. Gasper, Linearization of the product of Jacobi polyn omials I, Canad. J. Math. 22 (1970) 171–175

work page 1970

[23] [23]

Gonc ¸alves, G

F. Gonc ¸alves, G. V edana, Sphere packings in Euclideanspace with forbidden distances, arXiv:2308.03925

work page arXiv

[24] [24]

Hardin, E

D. Hardin, E. Saff, Discretizing manifolds via minimum energy points, Notices of the American Math. Soc. , Nov. 2004, 1186–1194

work page 2004

[25] [25]

G. A. Kabatianskii, V . I. Levenshtein, Bounds for packi ngs on a sphere and in space, Probl. Inform. Transm. 14 (1978) 1–17

work page 1978

[26] [26]

A. V . Kolushov, V . A. Y udin, Extremal dispositions of points on the sphere, Anal. Math. 23 (1997) 25–34

work page 1997

[27] [27]

V . I. Levenshtein, Universal bounds for codes and desig ns, in Handbook of Coding Theory, V . S. Pless and W. C. Huffman, Eds., Elsevier, Amsterdam, Ch. 6, 499–648 (1998)

work page 1998

[28] [28]

Nebe, A fourth extremal even unimodular lattice of di mension 48, Discr

G. Nebe, A fourth extremal even unimodular lattice of di mension 48, Discr . Math.331 (2014) 133–136

work page 2014

[29] [29]

Niemeier, Deﬁnite quadratische Formen der Dimen sion 24 und Diskriminante I, J

H.-V . Niemeier, Deﬁnite quadratische Formen der Dimen sion 24 und Diskriminante I, J. Numb. Th. 5 (1973) 142–178

work page 1973

[30] [30]

Saff, A.B.J

E.B. Saff, A.B.J. Kuijlaars, Distributing many points on a sphere, The Mathematical Intelligencer, 19 (1997) 5–11

work page 1997

[31] [31]

R. E. Schwartz, Five Point Energy Minimization: A Synop sis, Constr. Approx. 51 (2020) 537–564

work page 2020

[32] [32]

I. J. Schoenberg, Positive deﬁnite functions on sphere s, Duke Math. J. 9 (1942) 96–108

work page 1942

[33] [33]

Viazovska, The sphere packing problem in dimension 8 , Ann

M. Viazovska, The sphere packing problem in dimension 8 , Ann. of Math. 185 (2017) 991–1015. INSTITUTE OF MATHEMATICS AND INFORMATICS , B ULGARIAN ACADEMY OF SCIENCES , 8 G B ONCHEV STR ., 1113 S OFIA , B ULGARIA Email address: peter@math.bas.bg INSTITUTE OF MATHEMATICS AND INFORMATICS , B ULGARIAN ACADEMY OF SCIENCES , 8 G B ONCHEV STR ., 1113 S OFIA , B...

work page 2017