Here is a list of self-dual objects from classical coding theory — codes, lattices, and polytopes.
| Code | Description |
|---|---|
| 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. |
| 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. |
| Extended quaternary Golay code | An extended quadratic residue quaternary linear \((24,4^{12},12)_{\mathbb{Z}_4}\) code that is a quaternary version of the Golay code. The code has Lee distance 12, Hamming distance 8, and Euclidean distance 16 [1]. The code maps to a binary code under the Gray map whose distance distribution is equal to the MacWilliams transform of the distance distribution of its dual code. |
| Harada-Kitazume code | A member of a family of extremal Type II self-dual codes over \(\mathbb{Z}_4\) that yield all Niemeier lattices via Construction \(A_4\). |
| 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. Two copies of the code yield the \((6,54,1)\) double Hessian polyhedron (a.k.a. diplo-Schläfli) code. The code can be obtained from the Schläfli graph [2; Ch. 9]. The (antipodal pairs of) points of the (double) Hessian polyhedron correspond to the 27 lines on a smooth cubic surface in the complex projective plane [3–7]. |
| Klemm code | A member of a family of self-dual linear \((4m,4^1 2^{4m-2})_{\mathbb{Z}_4}\) codes. Its generator matrix consists of a sum of the generator matrix of the repetition code and twice the generator matrix of the SPC code [8]. |
| Niemeier lattice | One of the 24 positive-definite even unimodular lattices of rank 24. |
| Octacode | The unique self-dual linear \((8,4^4,6)_{\mathbb{Z}_4}\) code of Euclidean distance 8. Its shortened version is called the \((7,4^3,6)_{\mathbb{Z}_4}\) heptacode. |
| 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 (\(q=6\)). |
| Pseudo Golay code | Any one of 13 quaternary extremal Type II self-dual linear codes over \(\mathbb{Z}_4\) of length 24 whose mod-two reduction (mapping \(0,1,2,3\) to \(0,1,0,1\)) is the Golay code [9; Thm. 11]. Each code has Lee distance 12, Hamming distance 8, and Euclidean distance 16 [9; Thm. 9]. |
| Quadrature PSK (QPSK) code | A quaternary encoding into a constellation of four points distributed equidistantly on a circle. For the case of \(\pi/4\)-QPSK, the constellation is \(\{e^{\pm i\frac{\pi}{4}},e^{\pm i\frac{3\pi}{4}}\}\). |
| Self-dual additive code | An additive \((n,2^n)_q\) code that is equal to its dual, \(C^\perp = C\), where the dual is defined with respect to some inner product, usually the trace-Hermitian inner product. |
| Self-dual code over \(R\) | An additive linear code \(C\) over a ring \(R\) that is equal to its dual, \(C^\perp = C\), where the dual is defined with respect to some inner product. |
| Self-dual code over \(\mathbb{Z}_4\) | An additive linear code \(C\) over \(\mathbb{Z}_4\) that is equal to its dual, \(C^\perp = C\), where the dual is defined with respect to the standard inner product. The code contains \(2^n\) codewords [10; Corr. 1.3]. |
| Self-dual code over \(\mathbb{Z}_q\) | An additive linear code \(C\) over \(\mathbb{Z}_q\) that is equal to its dual, \(C^\perp = C\), where the dual is defined with respect to the standard inner product. |
| Self-dual linear code | An \([n,n/2]_q\) code that is equal to its dual, \(C^\perp = C\), where the dual is defined with respect to an inner product, most commonly either Euclidean or Hermitian. Self-dual codes exist only for even lengths and have dimension \(k=n/2\). A code that is equivalent to its dual is called iso-dual. |
| Self-dual polytope code | A spherical code whose codewords are the vertices of a self-dual polytope. |
| 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. Simplex spherical codewords in 2 (3, 4) dimensions form the vertices of a triangle (tetrahedron, 5-cell) In general, the code makes up the vertices of an \(n\)-simplex. The union of a simplex and its antipodal simplex forms the vertices of a bi-simplex, which has \(2(n+1)\) vertices. |
| Unimodular lattice | A lattice that is equal to its dual, \(L^\perp = L\). Unimodular lattices have \(\det L = \pm 1\). |
| 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. The code is optimal and unique up to equivalence [11–13]. Antipodal pairs of points of the \(4_{21}\) real polytope code correspond to the 120 tritangent planes of a canonical sextic curve in \(\mathbb{C}P^3\) [4–7]. |
| \((12,4^6,6)_4\) Dodecacode | The unique trace-Hermitian self-dual additive \((12,4^6,6)_4\) code. Its codewords are cyclic permutations of \((\omega 10100100101)\), where \(\mathbb{F}_4=\{0,1,\omega,\bar{\omega}\}\) is the quaternary Galois field [14; Sec. 2.4.8]. Another generator matrix can be found in [15; Exam. 9.10.8]. |
| \(C_{m,r}\) code | A member of a family of Type IV self-dual quaternary linear codes over \(\mathbb{Z}_4\) generated by \(\textnormal{RM}(r,m) + 2\textnormal{RM}(m-r-1,m)\) for \(3r \leq m-1\) [8]. |
| \(E_8\) Gosset lattice | Unimodular even BW lattice in dimension \(8\), consisting of the Cayley integral octonions rescaled by \(\sqrt{2}\). The lattice corresponds to the \([8,4,4]\) Hamming code via Construction A. |
| \([24, 12, 8]\) Extended Golay code | A self-dual \([24, 12, 8]\) code that is obtained from the Golay code by adding a parity check. Up to equivalence, it is unique for its parameters [16]. |
| \([2^m,2^m-m-1,4]\) Extended Hamming code | Member of an infinite family of RM\((m-2,m)\) codes with parameters \([2^m,2^m-m-1, 4]\) for \(m \geq 2\) that are extensions of the Hamming codes by a parity-check bit. |
| \([2m+2,m+1]\) Karlin double circulant code | Member of the family of \([2m+2,m+1]\) double circulant codes such that \(m\) is prime of the form \(8k+3\) for some \(k\), and \(2m+2\) is a multiple of eight. See [17; Ch. 16] for their generator matrix. Karlin double circulant codes can be mapped to extended cyclic and extended quadratic-residue codes over \(\mathbb{F}_4\) [18,19][17; Ch. 16][14; Sec. 2.4.2] by identifying \((0,\omega,\bar{\omega},1)\) with \((00),(10),(01),(11)\) [18]. |
| \([2q+2,q+1]_3\) Pless symmetry code | A member of a family of self-dual ternary \([2q+2,q+1]_3\) codes for any power of an odd prime satisfying \(q \equiv 2\) modulo 3. |
| \([4,2,2]_{\mathbb{Z}_6}\) senary code | A self-dual code over \(\mathbb{Z}_6\) that is one of two such codes, up to permutations [20]. |
| \([4,2,3]_3\) Tetracode | The \([4,2,3]_3\) ternary self-dual MDS code that has connections to lattices [12]. |
| \([4,2,3]_4\) RS\(_4\) code | A Type II Euclidean self-dual RS code that is the smallest quaternary extended QR code [14; Sec. 2.4.2]. |
| \([48,24,12]\) self-dual code | An extended quadratic-residue code that is known to be the only self-dual doubly even code at its parameters [21]. |
| \([6,3,4]_4\) Hexacode | The \([6,3,4]_4\) self-dual MDS code that has connections to projective geometry, lattices [12], and conformal field theory [22]. |
| \([8,4,4]\) extended Hamming code | Extension of the \([7,4,3]\) Hamming code by a parity-check bit. The smallest doubly even self-dual code. |
| \(\Lambda_{24}\) Leech lattice | Even unimodular lattice in 24 dimensions that exhibits optimal packing. Its automorphism group is the Conway group \(.0\) a.k.a. Co\(_0\). |
| \(\mathbb{Z}^n\) hypercubic lattice | Lattice-based code consisting of all integer vectors in \(n\) dimensions. Its generator matrix is the \(n\)-dimensional identity matrix. Its automorphism group consists of all coordinate permutations and sign changes. |
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