Welcome to the Spherical Kingdom.

Constant-energy code
Code whose codewords are points on a complex or real hypersphere in either whose radius squared is called the energy. Typically, only angular distances between points are relevant for code performance, so one can normalize codewords of a constant-energy code to obtain up a spherical code, i.e., a constant energy code with energy one.
Parents:
Error-correcting code (ECC).
Parent of:
Spherical code.
Cousins:
Sphere packing, Constant-excitation (CE) code.

Spherical code
Code whose codewords are points on an \(n\)-dimensional sphere \(S^{n}\) with radius one. It is denoted as \((n,M,\rho)\), where \(n\) is the code dimension, \(M\) is the size or number of codewords, and \(\rho\) is the squared minimum distance, i.e., the smallest Euclidean distance between pairs of distinct codewords, \begin{align}
\rho=\min\left\{ \left\Vert x-y\right\Vert ^{2}\,\text{s.t.}\,x,y\in C,\,\,x\neq y\right\}~.
\end{align}
Protection: The Euclidean distance between two points is related to the dot product as \begin{align}
\left\Vert x-y\right\Vert^{2} = 2-2x \cdot y~,
\end{align} where \(x\cdot y\) is the Euclidean inner product. As a result, the angular distance, \begin{align}
\theta=\arccos(x\cdot y) \in[0,\pi]~,
\end{align} can be equivalently used to quantify code performance.
Parents:
Constant-energy code.
Parent of:
Annealing-based spherical code, Binary balanced spherical code, Laminated spherical code, Lattice-shell code, Polytope code, Slepian group-orbit code, Wrapped spherical code.

Slepian group-orbit code[1]
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 group \(O(n)\) of rotations in \(n\) dimensions. Neither the vector nor the group are unique for a given code.
Protection: Code properties depend on the relationship between the group and the initial vector, and the number of codewords is the number of cosets of an initial vector's symmetry subgroup in \(G\) per the orbit-stabilizer theorem. See Refs. [2][3][4][5] for allowed code parameters.
Parents:
Spherical code, Group-orbit code.
Parent of:
Permutation spherical code, Real-Clifford subgroup-orbit code, Torus-layer spherical code (TLSC).
Cousins:
Linear binary code, Binary antipodal code, Linear code over \(G\).
Cousin of:
Polytope code.

Annealing-based spherical code[6][7][8]
Code whose codewords are obtained from a simulated annealing or energy-repulsion numerical optimization procedure.
Parents:
Spherical code.

Binary balanced spherical code
An \((n-1,K,\frac{nd}{nw-w^2})\) spherical code obtained from a constant-weight-\(w\) binary \((n,K,d)\) code via a component-wise binary balanced mapping (also known as the CW\(_2\) construction), \begin{align}
\begin{split}
0&\to\sqrt{\frac{w}{n\left(n-w\right)}}\\1&\to -\sqrt{\frac{n-w}{nw}}~.
\end{split}
\end{align} This construction can be extended to the general balanced binary construction CW\(_q\) for spherical code alphabets of size \(q\) [9; Sec. 6.6].
Parents:
Spherical code, Concatenated code.
Cousins:
Constant-weight code, Binary PSK (BPSK) code.

Laminated spherical code[10]
Spherical code whose codewords are obtained from a recursive procedure that is similar to the procedure that creates laminated lattices.
Parents:
Spherical code.

Lattice-shell code[11]
Spherical code whose codewords are scaled versions of points on a lattice. 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.
Parents:
Spherical code.
Parent of:
Cubeoctahedron code, \(BW_{32}\) lattice-shell code, \(D_4\) lattice-shell code, \(E_6\) lattice-shell code, \(E_8\) Gosset lattice-shell code, \(\Lambda_{16}\) lattice-shell code, \(\Lambda_{24}\) Leech lattice-shell code.
Cousins:
Lattice-based code, Cyclic code.

Polytope code
Spherical code whose codewords are the vertices of a polytope, i.e., a geometrical figure bounded by lines, planes, and hyperplanes [12]. Polytopes in two (three) real or complex dimensions are called polygons (polyhedra).
Parents:
Spherical code.
Parent of:
120-cell code, Biorthogonal spherical code, Cubeoctahedron code, Hessian polyhedron code, Icosahedron code, Rectified Hessian polyhedron code, Simplex spherical code, Snub-cube code, Square-antiprism code, Witting polytope code.
Cousins:
Slepian group-orbit code.

Wrapped spherical code[13]
Spherical code in dimension \(n\) whose codewords are obtained from centers of spheres from a finite \(S^{n-1}\)-sphere packing of \(\mathbb{R}^{n}\) that is "wrapped" onto \(S^n\).
Parents:
Spherical code.

Permutation spherical code[14][15]
Slepian group-orbit code whose codewords are constructed from an arbitrary unit vector in two possible variants. Variant 1 consists of codewords that are permutations of the vector's coordinates, while Variant 2 consists of such permutations and all possible sign changes of the vector's components.
Parents:
Slepian group-orbit code.
Parent of:
Biorthogonal spherical code, Simplex spherical code.

Real-Clifford subgroup-orbit code[16][17]
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 [18], 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 \(\{4,8,16\}\).
Parents:
Slepian group-orbit code.
Parent of:
Witting polytope code, \(24\)-cell code, \(BW_{32}\) lattice-shell code, \(\Lambda_{16}\) lattice-shell code.
Cousins:
Barnes-Wall (BW) lattice code.
Cousin of:
Disphenoidal \(288\)-cell code.

Torus-layer spherical code (TLSC)[19]
Code whose codewords are elements of a foliation of the \(2n-1\)-dimensional hypersphere \(S^{2n-1}\) using flat tori \(S^1\times S^1\cdots\times S^1\). Related constructions include the spherical codes by Hopf foliations (SCHF) [20].
Parents:
Slepian group-orbit code.
Parent of:
Polyphase code.

Polyphase code[21][22][23][24][25][26][27][28][29][30][31]
A spherical code obtained from a binary code, \(q\)-ary code, or \(q\)-ary code over \(\mathbb{Z}_q\) via a component-wise mapping of each \(q\)-ary digit to a \(q\)th root of unity.
Parents:
Torus-layer spherical code (TLSC), Concatenated code.
Parent of:
Binary antipodal code, Phase-shift keyring (PSK) code.
Cousins:
Galois-field \(q\)-ary code, \(q\)-ary code over \(\mathbb{Z}_q\), Simplex spherical code, Biorthogonal spherical code.

Phase-shift keyring (PSK) code
A \(q\)-ary phase-shift keyring (\(q\)-PSK) encodes one \(q\)-ary digit of information into a constellation of \(q\) points distributed equidistantly on a circle in \(\mathbb{C}\) or, equivalently, \(\mathbb{R}^2\). For example, such a constellation could be \begin{align}
\{1,e^{i\frac{2\pi}{q}},\cdots,e^{i\frac{2\pi}{q}(q-1)}\}~.
\end{align} Each point is typically associated with a complex amplitude of an electromagnetic signal, and information is encoded into the phase of that signal.
Parents:
Polyphase code, Quadrature-amplitude modulation (QAM) code.
Parent of:
Binary PSK (BPSK) code, Quadrature PSK (QPSK) code.
Cousins:
Gray code, Cat code.
Cousin of:
PSK c-q code.

Cubeoctahedron code
Spherical \((3,12,1)\) code whose codewords are the vertices of the cubeoctahedron. Codewords form the minimal lattice-shell code of the \(D_3\) face-centered cubic (fcc) lattice.
Protection: Code yields an optimal solution to the kissing problem in 3D [32].
Parents:
Polytope code, Lattice-shell code.
Cousins:
\(D_3\) face-centered cubic (fcc) lattice code.

\(BW_{32}\) lattice-shell code
Spherical code whose codewords are points on the \(BW_{32}\) Barnes-Wall lattice normalized to lie on the unit sphere.
Parents:
Lattice-shell code, Real-Clifford subgroup-orbit code.
Cousins:
\(BW_{32}\) Barnes-Wall lattice code.

\(D_4\) lattice-shell code
Spherical code whose codewords are points on the \(D_4\) lattice normalized to lie on the unit sphere.
Parents:
Lattice-shell code.
Parent of:
Disphenoidal \(288\)-cell code.
Cousins:
\(D_4\) hyper-diamond lattice code.

\(E_6\) lattice-shell code
Spherical code whose codewords are points on the \(E_6\) lattice normalized to lie on the unit sphere.
Parents:
Lattice-shell code.
Parent of:
Rectified Hessian polyhedron code.
Cousins:
\(E_6\) root lattice code.

\(E_8\) Gosset lattice-shell code
Spherical code whose codewords are points on the \(E_8\) Gosset lattice normalized to lie on the unit sphere.
Protection: Smallest-shell code yields an optimal solution to the kissing problem in 8D [33][34].
Parents:
Lattice-shell code.
Parent of:
Witting polytope code.
Cousins:
\(E_8\) Gosset lattice code.

\(\Lambda_{16}\) lattice-shell code
Spherical code whose codewords are points on the \(\Lambda_{16}\) Barnes-Wall lattice normalized to lie on the unit sphere.
Parents:
Lattice-shell code, Real-Clifford subgroup-orbit code.
Cousins:
\(\Lambda_{16}\) Barnes-Wall lattice code.

\(\Lambda_{24}\) Leech lattice-shell code
Spherical code whose codewords are points on the \(\Lambda_{24}\) Leech lattice normalized to lie on the unit sphere.
Protection: Smallest-shell code yields an optimal solution to the kissing problem in 24D [34].
Parents:
Lattice-shell code.
Cousins:
\(\Lambda_{24}\) Leech lattice code.

\(24\)-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 code forms a spherical 5-design [35]. A realization of the codewords consists of the 24 permutations of the four vectors \((0,0,\pm 1,\pm 1)\); see [36; Table 3] for another realization. A realization in terms of quaternions yields the 24 elements of the binary tetrahedral group \(2T\) [37].
Protection: Code yields an optimal solution to the kissing problem in 4D [38][39].
Parents:
600-cell code, Disphenoidal \(288\)-cell code, Real-Clifford subgroup-orbit code.
Cousin of:
Biorthogonal spherical code.

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. The code forms a spherical 5-design [40]. See [41; pg. 127][42; pg. 126] for realizations of the 72 codewords.
Parents:
Polytope code, \(E_6\) lattice-shell code.
Cousins:
Hessian polyhedron code.

Witting polytope code
Spherical \((8,240,1)\) code whose codewords are the vertices of the Witting complex polytope and the \(4_{21}\) real polytope. This code is optimal and unique up to equivalence [34][42] and forms a tight spherical 7-design [42; Ch. 14]. Codewords form the minimal lattice-shell code of the \(E_8\) lattice.
Protection: Code yields an optimal solution to the kissing problem in 4D [38][39].
Parents:
Polytope code, \(E_8\) Gosset lattice-shell code, Real-Clifford subgroup-orbit code.

120-cell code
Spherical \((4,600,(7-3\sqrt{5})/4)\) code whose codewords are the vertices of the 120-cell. The code forms a spherical 11-design because its vertices can be divided into five 600-cells, each of which forms said design. See [43; Table 1][12][36; Table 3] for realizations of the 600 codewords.
Parents:
Polytope code.
Parent of:
600-cell code.

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. Biorthogonal spherical codes are the only tight spherical 3-designs [9; Tab. 9.3].
Protection: Biorthogonal spherical codes saturate the third Rankin bound [9].
Parents:
Permutation spherical code, Polytope code.
Parent of:
Quadrature PSK (QPSK) code.
Cousins:
Binary antipodal code, \(24\)-cell code.
Cousin of:
Hadamard code, Polyphase code, Reed-Muller (RM) code.

Hessian polyhedron code
Spherical \((6,27,3/2)\) code whose codewords are the vertices of the Hessian complex polyhedron and the \(2_{21}\) real polytope. The code forms a tight spherical 4-design [44; Exam. 7.3]. See [9; Exam. 1.2.5] ([41; pg. 119]) for a real (complex) realization of the 27 codewords.
Parents:
Polytope code.
Cousins:
\(E_6\) root lattice code.
Cousin of:
Rectified Hessian polyhedron code.

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 code forms a unique tight spherical 5-design [45][9; Ex. 9.6.1].
Protection: Optimal configuration of 12 points in 3D space [9; pg. 76].
Parents:
Polytope code.
Cousins:
Golay code.
Cousin of:
Simplex spherical code.

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 [9; Sec. 7.7] for a parameterization.
Protection: Simplex spherical codes for \(2 < \rho \leq 4\) saturate the first two Rankin bounds [9].
Parents:
Permutation spherical code, Polytope code.
Cousins:
Binary antipodal code, Icosahedron code.
Cousin of:
Hadamard code, Polyphase code, Simplex code.

Snub-cube code
Spherical \((3,24,0.55384)\) code whose codewords are the vertices of the snub cube.
Protection: Optimal configuration of 24 points in 3D space [9; pg. 78].
Parents:
Polytope code.

Square-antiprism code
Spherical \((3,8,4(4-\sqrt{2})/7)\) code whose codewords are the vertices of the square antiprism.
Protection: Optimal configuration of nine points in 3D space [9; pg. 73].
Parents:
Polytope code.

Quadrature PSK (QPSK) code[46]
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 \(\{e^{\pm i\frac{\pi}{4}},e^{\pm i\frac{3\pi}{4}}\}\).
Parents:
Phase-shift keyring (PSK) code, Biorthogonal spherical code.

Binary antipodal code
Also known as a binary signal constellation. An \((n,K,4d/n)\) spherical code obtained from a binary \((n,K,d)\) code via a component-wise antipodal mapping (also known as a Euclidean-space image) \(0\to +1\) and \(1 \to -1\) [9; Example 1.2.1].
Parents:
Polyphase code.
Parent of:
Binary PSK (BPSK) code.
Cousins:
Binary PSK (BPSK) code, Binary code.
Cousin of:
Biorthogonal spherical code, Simplex spherical code, Slepian group-orbit code.

Binary PSK (BPSK) code[47]
Also called a binary antipodal modulation, phase-reversal keyring (PRK), or antipodal signaling. 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 in a PAM, PSK, or QAM scheme.
Parents:
Phase-shift keyring (PSK) code, Binary antipodal code.
Cousins:
Pulse-amplitude modulation (PAM) code, Linear binary code, Two-component cat code, Polar c-q code.
Cousin of:
BPSK c-q code, Binary antipodal code, Binary balanced spherical code.

Disphenoidal \(288\)-cell code
Spherical \((4,48,2-\sqrt{2})\) code [9; 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 \(2O\) [37; Sec. 8.6].
Parents:
\(D_4\) lattice-shell code.
Parent of:
\(24\)-cell code.
Cousins:
Real-Clifford subgroup-orbit code.

600-cell code
Spherical \((4,120,(3-\sqrt{5})/2)\) code whose codewords are the vertices of the 600-cell. The code forms a spherical 11-design that is unique up to equivalence [48]. See [49; Table 1][36; Table 3] for realizations of the 120 codewords. A realization in terms of quaternions yields the 120 elements of the binary icosahedral group \(2I\) [37].
Parents:
120-cell code.
Parent of:
\(24\)-cell code.

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