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).
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. Cousins: Linear 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. Cousin of: Niset-Andersen-Cerf 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. [3][4][5][6] for allowed code parameters. Parents: Spherical code, Group-orbit code. 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.
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.
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 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.
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.
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.
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. 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.
$$E_6$$ lattice code Lattice in dimension $$6$$. Parents: Lattice-based code.
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. Cousin of: Binary PSK (BPSK) code.
Code whose codewords are obtained from a simulated annealing or energy-repulsion numerical optimization procedure. Parents: Spherical code.
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. 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. 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.
$$BW_{32}$$ 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. 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]. 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.
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. 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. Cousin of: Polyphase 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 $$1 \to -1$$ [7; 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.
BW lattice in dimension $$32$$. Parents: Barnes-Wall (BW) lattice code. Cousin of: $$BW_{32}$$ lattice-shell 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 $$\sqrt{2}$$. Cousin of: $$D_4$$ lattice-shell code.
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 $$2$$. Cousins: Hamming code, Octacode. Cousin of: $$E_8$$ Gosset lattice-shell code.
BW lattice in dimension $$16$$. Parents: Barnes-Wall (BW) lattice code. Cousin of: $$\Lambda_{16}$$ lattice-shell code.
A positive-definite even unimodular lattice of rank 24. Parents: Unimodular lattice code. Parent of: $$\Lambda_{24}$$ Leech lattice 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 [42]. Parents: Polytope code, Lattice-shell 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].
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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