Here is a list of codes related to spherical designs.

Code | Description | Relation |
---|---|---|

120-cell code | Spherical \((4,600,(7-3\sqrt{5})/4)\) code whose codewords are the vertices of the 120-cell. See [2][1; Table 1][3; Table 3] for realizations of the 600 codewords. | The code forms a spherical 11-design because its vertices can be divided into five 600-cells, each of which forms said design. |

24-cell code | Spherical \((4,24,1)\) code whose codewords are the vertices of the 24-cell. Codewords form the minimal lattice-shell code of the \(D_4\) lattice. | The 24-cell code is a spherical 5-design [4]. |

600-cell code | Spherical \((4,120,(3-\sqrt{5})/2)\) code whose codewords are the vertices of the 600-cell. See [5; Table 1][3; Table 3] for realizations of the 120 codewords. A realization in terms of quaternions yields the 120 elements of the binary icosahedral group \(2I\) [6]. | The 600-cell code forms a spherical 11-design that is unique up to equivalence [7]. |

Binary PSK (BPSK) code | Encodes one bit of information into a constellation of antipodal points \(\pm\alpha\) for complex \(\alpha\). These points are typically associated with two phases of an electromagnetic signal. | |

Biorthogonal spherical code | Spherical \((n,2n,2)\) code whose codewords are all permutations of the \(n\)-dimensional vectors \((0,0,\cdots,0,\pm1)\), up to normalization. The code makes up the vertices of an \(n\)-orthoplex (a.k.a. hyperoctahedron or cross polytope). | Biorthogonal spherical codes are the only tight spherical 3-designs [8; Tab. 9.3]. |

Cameron-Goethals-Seidel (CGS) isotropic subspace code | Member of a \((q(q^2-q+1),(q+1)(q^3+1),2-2/q^2)\) family of spherical codes for any prime-power \(q\). Constructed from generalized quadrangles, which in this case correspond to sets of totally isotropic points and lines in the projective space \(PG_{5}(q)\) [8; Ex. 9.4.5]. There exist multiple distinct spherical codes using this construction for \(q>3\) [9]. | |

Combinatorial design code | A constant-weight binary code that is mapped into a combinatorial \(t\)-design. | |

Disphenoidal 288-cell code | Spherical \((4,48,2-\sqrt{2})\) code [8; Ex. 1.2.6] whose codewords are the vertices of the disphenoidal 288-cell. Codewords are the union of two 24-point lattice shells of the \(D_4\) lattice. The first shell consists of the 24 permutations of the four vectors \((0,0,\pm 1,\pm 1)\), and the second of the 16 vectors \((\pm 1,\pm 1,\pm 1,\pm 1)\) and the 8 permutations of the vectors \((0,0,0,\pm 2)\). A realization in terms of quaternions yields the 48 elements of the binary octahedral group \(2O\) [6; Sec. 8.6]. | The disphenoidal 288-cell code forms a spherical 7-design [10]. |

Golay code | A \([23, 12, 7]\) perfect binary linear code with connections to various areas of mathematics, e.g., lattices [11] and sporadic simple groups [12]. Adding a parity bit to the code results in the \([24, 12, 8]\) extended Golay code. Up to equivalence, both codes are unique for their respective parameters [13]. Shortening the Golay code yields the \([22,10,8]\), \([22,11,7]\), and \([22,12,6]\) shortened Golay codes [14]. The dual of the Golay code is its \([23,11,8]\) even-weight subcode [15,16]. | The dual of the Golay code forms a spherical three-design under the antipodal mapping [17; Exam. 9.3]. |

Hessian polyhedron code | Also known as the Schlafli configuration. Spherical \((6,27,3/2)\) code whose codewords are the vertices of the Hessian complex polyhedron and the \(2_{21}\) real polytope. Two copies of the code yield the \((6,54,1)\) double Hessian polyhedron code. | The Hessian polytope code forms a tight spherical 4-design [18; Exam. 7.3]. |

Hexacode | The \([6,3,4]_4\) self-dual MDS code that has connections to projective geometry, lattices [11], and conformal field theory [19]. Puncturing the code yields the perfect \([5,3,3]_4\) quaternary Hamming code known as the shortened hexacode or shorter hexacode [20]. Both codes are sometimes refereed to as Golay codes over \(GF(4)\). | The hexacode is a complex spherical 3-design [21]. |

Icosahedron code | Spherical \((3,12,2-2/\sqrt{5})\) code whose codewords are the vertices of the icosahedron (alternatively, the centers of the faces of a dodecahedron, the icosahedron's dual polytope). | The icosahedron code forms a unique tight spherical 5-design [17][8; Ex. 9.6.1]. |

Kerdock spherical code | Family of \((n=2^{2r},n^2,2-2/\sqrt{n})\) spherical codes for \(r \geq 2\), obtained from Kerdock codes via the antipodal mapping [8; pg. 157]. These codes are optimal for their parameters for \(2\leq r\leq 5\), they are unique for \(r\in\{2,3\}\), and they form spherical 3-designs because their codewords are unions of \(2^{2r-1}+1\) cross polytopes [22]. | Kerdock codes form spherical 3-designs because their codewords are unions of \(2^{2r-1}+1\) cross polytopes [22]. |

Lattice-shell code | Spherical code whose codewords are scaled versions of points on a lattice. A \(m\)-shell code consists of normalized lattice vectors \(x\) with squared norm \(\|x\|^2 = m\). Each code is constructed by normalizing a set of lattice vectors in one or more shells, i.e., sets of lattice points lying on a hypersphere. | Nonempty \(2m\)-shell codes of extremal even unimodular lattices in \(n\) dimensions form spherical \(t\)-designs with \(t=11\) (\(t=7\), \(t=3\)) if \(n \equiv 0\) (\(n \equiv 8\), \(n\equiv 16\)) modulo 24 [23,24]. Shells of \(A_n\) and \(D_n\) lattices form infinite families of spherical 3-designs [25; Exam. 2.9]. |

Petersen spherical code | A \((4,10,1/6)\) spherical code whose codewords correspond to vertices of the Peterson graph. Its Gram matrix is constructed by putting \(-2/3\) whenever two vertices are adjacent in the graph, and \(1/6\) otherwise. The code is optimal for its parameters [26]. | The Peterson spherical code forms a spherical two-design [26]. |

Phase-shift keying (PSK) code | A \(q\)-ary phase-shift keying (\(q\)-PSK) encodes one \(q\)-ary digit of information into a constellation of \(q\) points distributed equidistantly on a circle in \(\mathbb{C}\) or, equivalently, \(\mathbb{R}^2\). | |

Polygon code | Spherical \((1,q,4\sin^2 \frac{\pi}{q})\) code for any \(q\geq1\) whose codewords are the vertices of a \(q\)-gon. Special cases include the line segment (\(q=2\)), triangle (\(q=3\)), square (\(q=4\)), pentagon (\(q=5\)), and hexagon (\(q=6\)). | A \(q\)-gon a tight spherical \(q-1\) design. |

Quadrature PSK (QPSK) code | Also known as quadriphase PSK, 4-PSK, or 4-QAM. Quaternary encoding into a constellation of four points distributed equidistantly on a circle. For the case of \(\pi/4\)-QPSK, the constellation is \(\{e^{\pm i\frac{\pi}{4}},e^{\pm i\frac{3\pi}{4}}\}\). | |

Real-Clifford subgroup-orbit code | Slepian group-orbit code of dimension \(2^r\), approximate asympotic size \(2.38 \cdot 2^{r(r+1)/2+1}\), and distance \(1\). Code is constructed by applying elements of an index-two subgroup of the real Clifford group, when taken as a subgroup of the orthogonal group [27], onto the vector \((1,0,0,\cdots,0)\). This group is the automorphism group of BW lattice, and the resulting codes coincide with the optimal spherical codes for dimensions \(\{4,8,16\}\). | The orbit of any point under the real Clifford subgroup is a spherical 7-design [28], and some are 11-designs [29]. |

Rectified Hessian polyhedron code | Spherical \((6,72,1)\) code whose codewords are the vertices of the rectified Hessian complex polyhedron and the \(1_{22}\) real polytope. Codewords form the minimal lattice-shell code of the \(E_6\) lattice. See [30; pg. 127][11; pg. 126] for realizations of the 72 codewords. | The rectified Hessian polyhedron code forms a spherical 5-design [31]. |

Simplex spherical code | Spherical \((n,n+1,2+2/n)\) code whose codewords are all permutations of the \(n+1\)-dimensional vector \((1,1,\cdots,1,-n)\), up to normalization, forming an \(n\)-simplex. Codewords are all equidistant and their components add up to zero. For example, the spherical simplex code in \(n=3\) makes up the vertices of a tetrahedron. In general, the code makes up the vertices of an \(n\)-simplex. See [8; Sec. 7.7] for a parameterization. | Simplex spherical codes are the only tight spherical 2-designs [8; Tab. 9.3]. |

Slepian group-orbit code | Spherical code in \(n\) dimensions whose codewords correspond to points in an orbit of some initial vector under a generating group \(G\), which is a subgroup of the orthogonal group \(O(n)\) of rotations in \(n\) dimensions, i.e., the automorphism group of spherical codes under the Euclidean distance. Neither the vector nor the group are unique for a given code. | Slepian group-orbit codes can form spherical designs [31,32]. Polynomial invariants of a discrete subgroup \(G\) of the orthogonal group can be used to determine the design strength of orbits of \(G\) [33]. Let \(t+1\) be the degree of the lowest-degree \(G\)-invariant polynomial that is not a polynomial in the norm \(\left\Vert x\right\Vert^2\). Then, any orbit under \(G\) forms a Slepian group-orbit code that is also a spherical \(t\)-design. |

Spherical design code | Spherical code whose codewords are uniformly distributed in a way that is useful for determining averages of polynomials over the real sphere. A spherical code is a spherical design of strength \(t\), i.e., a \(t\)-design, if the average of any polynomial of degree up to \(t\) over its codewords is equal to the average over the entire sphere. | |

Spherical sharp configuration | A spherical code that is a spherical design of strength \(2m-1\) for some \(m\) and that has \(m\) distances between distinct points. All known spherical sharp configrations are either obtained from the Leech or \(E_8\) lattice, certain regular polytopes, or are CGS isotropic subspace spherical codes [34; Table 1]. | Spherical sharp configurations are spherical designs of strength \(2m-1\) for some \(m\). |

Unimodular lattice code | A lattice that is equal to its dual, \(L^\perp = L\). Unimodular lattices have \(\det L = \pm 1\). | A union of \(t\) shells of self-dual lattices and their shadows form spherical \(t\)-designs [35]. |

Witting polytope code | Spherical \((8,240,1)\) code whose codewords are the vertices of the Witting complex polytope, the \(4_{21}\) real polytope, and the minimal lattice-shell code of the \(E_8\) lattice. Taking its kissing configuration yields the \((7,56,1/3)\) spherical code. Both codes are optimal and unique up to equivalence [11,36,37]. | The Witting polytope code forms a tight spherical 7-design, while the \((7,56,1/3)\) spherical code forms a 5-design [36][11; Ch. 14]. |

\(BW_{32}\) lattice-shell code | Spherical code whose codewords are points on the \(BW_{32}\) Barnes-Wall lattice normalized to lie on the unit sphere. | |

\(D_4\) lattice-shell code | Spherical code whose codewords are points on the \(D_4\) lattice normalized to lie on the unit sphere. | \(D_4\) \(2m\)-shell codes can form spherical designs [38]. |

\(\Lambda_{16}\) lattice-shell code | Spherical code whose codewords are points on the \(\Lambda_{16}\) Barnes-Wall lattice normalized to lie on the unit sphere. | |

\(\Lambda_{24}\) Leech lattice-shell code | Spherical code whose codewords are points on the \(\Lambda_{24}\) Leech lattice normalized to lie on the unit sphere. The minimal shell of the lattice yields the \((24,196560,1)\) code, and recursively taking their kissing configurations yields the \((23,4600,1/3)\) and \((22,891,1/4)\) spherical codes [17]; all codes are optimal and unique for their parameters [36,37]. | Smallest-shell \((24,196560,1)\) code is a tight and unique spherical 11-design [11; Ch. 3]. The \((23,4600,1/3)\) and \((22,891,1/4)\) spherical codes are spherical 7- and 5-designs, respectively [9,36,37,39][34; Table 1]. |

## References

- [1]
- M. Waegell and P. K. Aravind, “Parity Proofs of the Kochen–Specker Theorem Based on the 120-Cell”, Foundations of Physics 44, 1085 (2014) arXiv:1309.7530 DOI
- [2]
- H. S. M. Coxeter. Regular polytopes. Courier Corporation, 1973.
- [3]
- S. Mamone, G. Pileio, and M. H. Levitt, “Orientational Sampling Schemes Based on Four Dimensional Polytopes”, Symmetry 2, 1423 (2010) DOI
- [4]
- H. Cohn et al., “TheD\({}_{\text{4}}\)Root System Is Not Universally Optimal”, Experimental Mathematics 16, 313 (2007) arXiv:math/0607447 DOI
- [5]
- M. Waegell and P. K. Aravind, “Critical noncolorings of the 600-cell proving the Bell–Kochen–Specker theorem”, Journal of Physics A: Mathematical and Theoretical 43, 105304 (2010) arXiv:0911.2289 DOI
- [6]
- L. Rastanawi and G. Rote, “Towards a Geometric Understanding of the 4-Dimensional Point Groups”, (2022) arXiv:2205.04965
- [7]
- P. Boyvalenkov and D. Danev, “Uniqueness of the 120-point spherical 11-design in four dimensions”, Archiv der Mathematik 77, 360 (2001) DOI
- [8]
- T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
- [9]
- H. Cohn and A. Kumar, “Universally optimal distribution of points on spheres”, Journal of the American Mathematical Society 20, 99 (2006) arXiv:math/0607446 DOI
- [10]
- N. J. A. Sloane, R. H. Hardin, and P. Cara, “Spherical designs in four dimensions”, Proceedings 2003 IEEE Information Theory Workshop (Cat. No.03EX674) DOI
- [11]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
- [12]
- F. J. MacWilliams and N. J. A. Sloane. The theory of error correcting codes. Elsevier, 1977.
- [13]
- P. Delsarte and J. M. Goethals, “Unrestricted codes with the golay parameters are unique”, Discrete Mathematics 12, 211 (1975) DOI
- [14]
- J. H. Conway and N. J. A. Sloane, “Orbit and coset analysis of the Golay and related codes”, IEEE Transactions on Information Theory 36, 1038 (1990) DOI
- [15]
- W. Feit. Some remarks on weight functions of spaces over GF(2), unpublished (1972)
- [16]
- C. L. Mallows and N. J. A. Sloane, “Weight enumerators of self-orthogonal codes”, Discrete Mathematics 9, 391 (1974) DOI
- [17]
- P. Delsarte, J. M. Goethals, and J. J. Seidel, “Spherical codes and designs”, Geometriae Dedicata 6, 363 (1977) DOI
- [18]
- A. Roy and S. Suda, “Complex spherical designs and codes”, (2011) arXiv:1104.4692
- [19]
- J. A. Harvey and G. W. Moore, “Moonshine, superconformal symmetry, and quantum error correction”, Journal of High Energy Physics 2020, (2020) arXiv:2003.13700 DOI
- [20]
- G. Hoehn, “Self-dual Codes over the Kleinian Four Group”, (2000) arXiv:math/0005266
- [21]
- V. V. Albert, private communication, 2024.
- [22]
- H. Cohn, D. de Laat, and N. Leijenhorst, “Optimality of spherical codes via exact semidefinite programming bounds”, (2024) arXiv:2403.16874
- [23]
- B. B. Venkov. Even unimodular extremal lattices, algebraic geometry and its applications. Trudy Mat. Inst. Steklov., 165:43–48, 1984.
- [24]
- B. Venkov. Réseaux et designs sphériques. In Réseaux euclidiens, designs sphériques et formes modulaires, volume 37 of Monogr. Enseign. Math., pages 10–86. Enseignement Math., Geneva, 2001.
- [25]
- E. Bannai and E. Bannai, “A survey on spherical designs and algebraic combinatorics on spheres”, European Journal of Combinatorics 30, 1392 (2009) DOI
- [26]
- C. Bachoc and F. Vallentin, “Optimality and uniqueness of the (4,10,1/6) spherical code”, Journal of Combinatorial Theory, Series A 116, 195 (2009) arXiv:0708.3947 DOI
- [27]
- G. Nebe, E. M. Rains, and N. J. A. Sloane, “The invariants of the Clifford groups”, (2000) arXiv:math/0001038
- [28]
- V. M. Sidelnikov, Journal of Algebraic Combinatorics 10, 279 (1999) DOI
- [29]
- V. M. Sidelnikov, “Orbital spherical 11-designs in which the initial point is a root of an invariant polynomial”, Algebra i Analiz, 11:4 (1999), 183–203; St. Petersburg Math. J., 11:4 (2000), 673–686
- [30]
- H. S. M. Coxeter. Regular Complex Polytopes. Cambridge University Press, 1991.
- [31]
- P. de la Harpe and C. Pache, “Spherical designs and finite group representations (some results of E. Bannai)”, European Journal of Combinatorics 25, 213 (2004) DOI
- [32]
- E. BANNAI, “Spherical t-designs which are orbits of finite groups”, Journal of the Mathematical Society of Japan 36, (1984) DOI
- [33]
- S. L. Sobolev, “Cubature Formulas on the Sphere Invariant under Finite Groups of Rotations”, Selected Works of S.L. Sobolev 461 DOI
- [34]
- H. Cohn, “Packing, coding, and ground states”, (2016) arXiv:1603.05202
- [35]
- C. Pache, “Shells of selfdual lattices viewed as spherical designs”, (2005) arXiv:math/0502313
- [36]
- E. Bannai and N. J. A. Sloane, “Uniqueness of Certain Spherical Codes”, Canadian Journal of Mathematics 33, 437 (1981) DOI
- [37]
- H. Cohn and A. Kumar, “Uniqueness of the (22,891,1/4) spherical code”, (2007) arXiv:math/0607448
- [38]
- M. Hirao, H. Nozaki, and K. Tasaka, “Spherical designs and modular forms of the \(D_4\) lattice”, (2023) arXiv:2303.09000
- [39]
- R. A. Wilson, “Vector stabilizers and subgroups of Leech lattice groups”, Journal of Algebra 127, 387 (1989) DOI