Welcome to the Analog Kingdom.

Analog code Encodes states (codewords) into coordinates in the \(n\)-dimensional real coordinate space \(\mathbb{R}^n\). The number of codewords may be infinite because the coordinate space is infinite-dimensional, so various restricted versions have to be constructed in practice. Protection: The primary application of analog codes is to transmist information using electromagnetic signals. The primary noise channel for such signals is the additive Gaussian white-noise channel (AGWN), which adds a random Gaussian-distributed displacement to each signal point. Protection of a constellation thus depends on how far apart its points are in terms of the Euclidean distance. As usual, there is a tradeoff between packing of space and level of protection. Parents: Error-correcting code (ECC). Parent of: Constant-energy code, Lattice-based code, Quadrature-amplitude modulation (QAM) code. Cousin of: Coherent-state constellation quantum code, Group-based code.
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: Analog code. Parent of: Spherical code. Cousins: Constant-excitation (CE) code.
Lattice-based code Encodes states (codewords) in coordinates of an \(n\)-dimensional lattice. The number of codewords may be infinite because the coordinate space is infinite-dimensional, so various restricted versions have to be constructed in practice. Since lattices are closed under addition, lattice-based codes can be thought of as linear codes over the reals. Protection: Lattices are characterized by the minimum (Euclidean) distance \(d_{\text{min}}\) between two lattice points, the kissing number \(K_{\text{min}}\) of nearest neighbors at each lattice point, and the volume \(V=\det G\), which is equal to the determinant of the lattice generator matrix \(G\). Parents: Analog code. Parent of: Antipode lattice code, Barnes-Wall (BW) lattice code, Hexagonal lattice code, Mod-2 lattice code, Unimodular lattice code, \(D_3\) face-centered cubic (fcc) lattice code, \(E_6\) lattice code, \(\Lambda_{24}\) Leech lattice code. Cousins: Linear code. Cousin of: Best code, Lattice-shell code, Linear code over \(G\), Multi-mode GKP code, Quadrature-amplitude modulation (QAM) code.
Quadrature-amplitude modulation (QAM) code Encodes into points into a subset of points lying on in \(\mathbb{R}^{2}\), here treated as \(\mathbb{C}\). Each pair of points is associated with a complex amplitude of an electromagnetic signal, and information is encoded into both the norm and phase of that signal [1; Ch. 16]. Parents: Analog code. Parent of: Phase-shift keyring (PSK) code, Pulse-amplitude modulation (PAM) code. Cousins: Lattice-based code, Gottesman-Kitaev-Preskill (GKP) code, Gray code. Cousin of: Niset-Andersen-Cerf code.
Slepian group-orbit code[2] 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. [3][4][5][6] for allowed code parameters. Parents: Spherical code, Group-orbit code. Parent of: Permutation spherical code, Sidelnikov group-orbit code, Torus-layer spherical code (TLSC). Cousins: Linear binary code, Binary antipodal code, Linear code over \(G\). Cousin of: Polytope 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.
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 \(q\) [7; Sec. 6.6]. Parents: Spherical code, Concatenated code. Cousins: Constant-weight code, Binary PSK (BPSK) code.
Permutation spherical code[8][9] 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.
Sidelnikov group-orbit code[10][11] 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 a particular family of orthogonal groups onto the vector \((1,0,0,\cdots,0)\). The code coincides with the optimal spherical codes for dimensions \(\{4,8,16\}\). Parents: Slepian group-orbit code. Parent of: \(BW_{32}\) lattice-shell code, \(D_4\) lattice-shell code, \(E_8\) Gosset lattice-shell code, \(\Lambda_{16}\) lattice-shell code.
Torus-layer spherical code (TLSC)[12] 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) [13]. Parents: Slepian group-orbit code. Parent of: Polyphase code.
Polyphase code[14][15][16][17][18][19][20][21][22][23][24] 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.
Antipode lattice code[25] Lattice code constructed via the antipode construction. Parents: Lattice-based code. Cousins: Anticode.
Barnes-Wall (BW) lattice code Member of a family of \(2^{m+1}\)-dimensional lattices, denoted as BW\(_{2^{m+1}}\), that are the densest lattices known. Members include the integer square lattice \(\mathbb{Z}^2\), \(D_4\), the Gosset \(E_8\) lattice, and the \(\Lambda_{16}\) lattice, corresponding to \(m\in\{0,1,2,3\}\), respectively. Protection: BW lattices in dimension \(2^{m+1}\) have a nominal coding gain of \(2^{m/2}\). Their kissing number is \(K_{\text{min}} = \prod_{i=1}^{m+1} (2^i + 2)\). Parents: Lattice-based code. Parent of: \(BW_{32}\) Barnes-Wall lattice code, \(D_4\) lattice code, \(E_8\) Gosset lattice code, \(\Lambda_{16}\) Barnes-Wall lattice code. Cousins: Reed-Muller (RM) code.
Hexagonal lattice code Two-dimensional lattice that exhibits optimal packing, solving the packing, kissing, covering and quantization problems. Its dual is the honeycomb lattice. Parents: Lattice-based code. Cousin of: Color code.
Mod-2 lattice code Lattice constructed from a binary linear \([n,k,d]\) code using Construction \(A_2\) [26]. Each binary codeword \(c\) of the code is mapped to an infinite set of points \(x\) such that \(x = c\) modulo two. Linearity of the code ensures that the resulting set of points forms a lattice. Parents: Lattice-based code. Parent of: \(D_4\) lattice code, \(E_8\) Gosset lattice code. Cousins: Linear binary code.
Unimodular lattice code Integral lattice of determinant \(\pm 1\). Protection: The minimum norm of a unimodular lattice satisfies \begin{align} \mu\leq2\left[\frac{n}{24}\right]+2 \end{align} unless \(n = 23\) [27]. Parents: Lattice-based code. Parent of: Niemeier lattice code, \(E_8\) Gosset lattice code. Cousins: Dual linear code.
\(\Lambda_{24}\) Leech lattice code[28] Lattice in 24 dimensions that exhibits optimal packing. Protection: The \(\Lambda_{24}\) lattice has a nominal coding gain of \(4\). Parents: Lattice-based code, Niemeier lattice code. Cousins: Golay code. Cousin of: \(\Lambda_{24}\) Leech lattice-shell code.
Pulse-amplitude modulation (PAM) code Encodes a \(q\)-ary digit into a constellation of equally spaced points on the real line. For example, a \(q\)-PAM scheme for \(q=8\) could encode the constellation \(\{ \pm \alpha,\pm 3\alpha,\pm 5\alpha, \pm 7\alpha \}\) with real scaling factor \(\alpha\). The points in the constellation are typically associated with one quadrature of an electromagnetic signal. Parents: Quadrature-amplitude modulation (QAM) code. Cousin of: Binary PSK (BPSK) code.
Annealing-based spherical code[29][30][31] Code whose codewords are obtained from a simulated annealing or energy-repulsion numerical optimization procedure. Parents: Spherical code.
Laminated spherical code[32] 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 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 [33]. 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[34] 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.
\(BW_{32}\) 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, Sidelnikov group-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, Sidelnikov group-orbit code. Parent of: \(24\)-cell code. Cousins: \(D_4\) 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 [35][36]. Parents: Lattice-shell code, Sidelnikov group-orbit 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, Sidelnikov group-orbit code. Cousins: \(\Lambda_{16}\) Barnes-Wall lattice 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 [7; Tab. 9.3]. Protection: Biorthogonal spherical codes saturate the third Rankin bound [7]. Parents: Permutation spherical code, Polytope code. Parent of: Quadrature PSK (QPSK) code. Cousins: Binary antipodal code, Reed-Muller (RM) code. Cousin of: Polyphase 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. 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 [7; Sec. 7.7] for a parameterization. Protection: Simplex spherical codes for \(2 < \rho \leq 4\) saturate the first two Rankin bounds [7]. Parents: Permutation spherical code, Polytope code. Cousins: Binary antipodal code, Simplex code, Icosahedron code. Cousin of: Polyphase code.
Quadrature PSK (QPSK) code[37] 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 \(1 \to -1\) [7; 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[38] 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, Quantum-classical polar code. Cousin of: Binary antipodal code, Binary balanced spherical code.
\(D_4\) lattice code BW lattice in dimension \(4\), which is the lattice corresponding to the \([4,1,4]\) repetition and \([4,3,2]\) SPC codes via the mod-two lattice construction. Protection: The \(D_4\) lattice has a nominal coding gain of \(\sqrt{2}\). Parents: Barnes-Wall (BW) lattice code, Mod-2 lattice code. Cousins: Repetition code, Single parity-check (SPC) code. Cousin of: \(D_4\) lattice-shell code.
\(E_8\) Gosset lattice code[40] BW lattice in dimension \(8\), which is the lattice corresponding to the \([8,4,4]\) Hamming code via the mod-two lattice construction. Protection: The \(E_8\) lattice has a nominal coding gain of \(2\). Parents: Barnes-Wall (BW) lattice code, Mod-2 lattice code, Unimodular lattice code. Cousins: Hamming code, Octacode. Cousin of: \(E_8\) Gosset lattice-shell code.
Niemeier lattice code[41] A positive-definite even unimodular lattice of rank 24. Parents: Unimodular lattice code. Parent of: \(\Lambda_{24}\) Leech lattice code. Cousins: Dual linear code, Quaternary code over \(\mathbb{Z}_4\), Octacode.
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 [42]. Parents: Polytope code, Lattice-shell code. Cousins: \(D_3\) face-centered cubic (fcc) 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: \(E_6\) 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 [36]. 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 [43]. A realization of the codewords consists of the 24 permutations of the four vectors \((0,0,\pm 1,\pm 1)\). Protection: Code yields an optimal solution to the kissing problem in 4D [44][45]. Parents: 600-cell code, \(D_4\) lattice-shell 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 [46]. See [47; pg. 127] for a realization 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 [36][26] and forms a tight spherical 7-design [26; 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 [44][45]. Parents: Polytope code, \(E_8\) Gosset lattice-shell code.
120-cell code Spherical \((4,600,0.0121407)\) code whose codewords are the vertices of the 120-cell. The code forms a spherical 11-design. See [48; Table 1][33] for a realization of the 600 codewords. Parents: Polytope code. Parent of: 600-cell 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 spherical 4-design. See [7; Exam. 1.2.5] ([47; pg. 119]) for a real (complex) realization of the 27 codewords. Parents: Polytope code. Cousins: \(E_6\) 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 [49][7; Ex. 9.6.1]. Protection: Optimal configuration of 12 points in 3D space [7; pg. 76]. Parents: Polytope code. Cousin of: Simplex spherical 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 [7; 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 [7; pg. 73]. Parents: Polytope 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 [50]. See [51; Table 1] for a realization of the 120 codewords. Parents: 120-cell code. Parent of: \(24\)-cell code.

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