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.
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.
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. Cousin of: Polytope code.
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.
Spherical code whose codewords are obtained from a recursive procedure that is similar to the procedure that creates laminated lattices. Parents: Spherical code.
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. 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. Cousins: Slepian group-orbit code.
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.
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.
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. Cousins: Barnes-Wall (BW) lattice code. Cousin of: Disphenoidal $$288$$-cell code.
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.
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.
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. 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.
$$BW_{32}$$ lattice-shell code Spherical code whose codewords are points on the $$BW_{32}$$ Barnes-Wall lattice normalized to lie on the unit sphere. 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.
$$\Lambda_{24}$$ 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]. 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].
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.
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 $$\{e^{\pm i\frac{\pi}{4}},e^{\pm i\frac{3\pi}{4}}\}$$.
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.
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.
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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