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Universal polar dual pairs of spherical codes found in $E_8$ and $Λ_{24}$

S. V. Borodachov, P. G. Boyvalenkov, P. D. Dragnev, D. P. Hardin, E. B. Saff, M. M. Stoyanova

TL;DR

This work identifies universal polar dual pairs of spherical codes arising from sharp codes embedded in the $E_8$ and Leech lattices, showing that for a broad class of absolutely monotone potentials $h$ the minima of $U_h(x,C)$ and $U_h(x,D)$ occur at the other code with equal normalized energy. The authors develop and apply a unified algebraic-lattice framework, employing the Smith normal form, derived codes, and Gauss–Jacobi quadrature to construct and certify maximal PULB-optimal pairs across $E_8$ and $\Lambda_{24}$, including numerous new configurations. They further extract several universal-optimal codes in real projective space $\mathbb{RP}^{21}$, notably a universally optimal antipodal code of size $1408$, by projecting Leech-layer structures and exploiting design-theoretic splits. The results bridge sharp-code design, discrete potential theory, and lattice geometry, with potential applications in quantum spherical codes and high-dimensional packing, while providing a broad, constructive catalog of universal polar dual pairs. A key methodological contribution is the extended derived-code theorem under hyperplane decompositions, enabling systematic generation of higher-strength designs from existing sharp codes.

Abstract

We identify universal polar dual pairs of spherical codes $C$ and $D$ such that for a large class of potential functions $h$ the minima of the discrete $h$-potential of $C$ on the sphere occur at the points of $D$ and vice versa. Moreover, the minimal values of their normalized potentials are equal. These codes arise from the known sharp codes embedded in the even unimodular extremal lattices $E_8$ and $Λ_{24}$ (Leech lattice). This embedding allows us to use the lattices' properties to find new universal polar dual pairs. In the process we extensively utilize the interplay between the binary Golay codes and the Leech lattice. As a byproduct of our analysis, we identify a new universally optimal (in the sense of energy) code in the projective space $\mathbb{RP}^{21}$ with $1408$ points (lines). Furthermore, we extend the Delsarte-Goethals-Seidel definition of derived codes from their seminal $1977$ paper and generalize their Theorem 8.2 to show that if a $τ$-design is enclosed in $k\leq τ$ parallel hyperplanes, then each of the hyperplane's sub-code is a $(τ+1-k)$-design in the ambient subspace.

Universal polar dual pairs of spherical codes found in $E_8$ and $Λ_{24}$

TL;DR

This work identifies universal polar dual pairs of spherical codes arising from sharp codes embedded in the and Leech lattices, showing that for a broad class of absolutely monotone potentials the minima of and occur at the other code with equal normalized energy. The authors develop and apply a unified algebraic-lattice framework, employing the Smith normal form, derived codes, and Gauss–Jacobi quadrature to construct and certify maximal PULB-optimal pairs across and , including numerous new configurations. They further extract several universal-optimal codes in real projective space , notably a universally optimal antipodal code of size , by projecting Leech-layer structures and exploiting design-theoretic splits. The results bridge sharp-code design, discrete potential theory, and lattice geometry, with potential applications in quantum spherical codes and high-dimensional packing, while providing a broad, constructive catalog of universal polar dual pairs. A key methodological contribution is the extended derived-code theorem under hyperplane decompositions, enabling systematic generation of higher-strength designs from existing sharp codes.

Abstract

We identify universal polar dual pairs of spherical codes and such that for a large class of potential functions the minima of the discrete -potential of on the sphere occur at the points of and vice versa. Moreover, the minimal values of their normalized potentials are equal. These codes arise from the known sharp codes embedded in the even unimodular extremal lattices and (Leech lattice). This embedding allows us to use the lattices' properties to find new universal polar dual pairs. In the process we extensively utilize the interplay between the binary Golay codes and the Leech lattice. As a byproduct of our analysis, we identify a new universally optimal (in the sense of energy) code in the projective space with points (lines). Furthermore, we extend the Delsarte-Goethals-Seidel definition of derived codes from their seminal paper and generalize their Theorem 8.2 to show that if a -design is enclosed in parallel hyperplanes, then each of the hyperplane's sub-code is a -design in the ambient subspace.
Paper Structure (24 sections, 33 theorems, 206 equations, 15 figures, 4 tables)

This paper contains 24 sections, 33 theorems, 206 equations, 15 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quadrature rules and spherical designs
  4. Delsarte-Goethals-Seidel bound, Levenshtein $1/N$-quadrature and bound on maximal codes, and Energy ULB
  5. Sharp codes, Energy ULB, and PULB
  6. The binary Golay code
  7. Lattices, generators, and the Smith normal form
  8. The $E_8$ and Leech ($\Lambda_{24}$) lattices
  9. Design strength of derived codes
  10. Maximal PULB-optimal pairs of codes embedded in the $E_8$ lattice
  11. The maximal PULB-optimal pair of the sharp code $C_{240}=(8,240,7)$ and its universal polar dual $C_{2160}=(8,2160,7)$
  12. The maximal PULB-optimal pair of the sharp code $C_{56}=(7,56,5)$ and its universal polar dual $C_{126}=(7,126,5)$.
  13. The maximal PULB-optimal pair generated by the sharp (Shläfli) code $C_{27}=(6,27,4)$ and $C_{72}=(6,72,5)$, the set of minimal vectors of $E_6$.
  14. The maximal PULB-optimal pair generated by the sharp code $C_{6}=(5,6,2)$ ($5$-dimensional simplex) and $C_{20}=(5,20,3)$ (equator of a six-dimensional cube).
  15. The maximal PULB-optimal pair generated by the Clebsch code $C_{16}=(5,16,3)$ and the cross-polytope $C_{10}=(5,10,3)$ in $\mathbb{R}^5$
  16. ...and 9 more sections

Key Result

Theorem 1.9

(BDHSS-JMAA, Bor-2) Suppose $C$ is a spherical $\tau$-design of cardinality $N$ on $\mathbb{S}^{n-1}$, where $\tau=:2k-1+\epsilon$, $\epsilon\in \{0,1\}$, and that the potential $h:[-1,1]\to (-\infty,\infty]$ is continuous on $[-1,1]$ (in extended sense), finite on $(-1,1)$, and has a (strictly) pos where the index set $I$, the quadrature nodes $\{\alpha_i\}_{i\in I}$, and the positive weights $\{

Figures (15)

  • Figure 1: The universal polar dual pair $(A,B)$ formed from projections of the first and second layers of the hexagonal lattice.
  • Figure 2: $E_8$ embedded $C_{240}$ and $C_{2160}$ PULB pair
  • Figure 3: $E_8$ embedded $C_{56}$ and $C_{126}$ PULB pairs
  • Figure 4: The PULB-optimal pair $(C_{54},C_{72})$ embedded in the $E_8$ lattice.
  • Figure 5: Leech lattice - the PULB-optimal pair $( \Lambda(2),\Lambda(3))$.
  • ...and 10 more figures

Theorems & Definitions (65)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Example 1.4
  • Example 1.5
  • Definition 1.6
  • Definition 1.7
  • Definition 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 55 more