Here is a list of codes related to lattices.

Code | Description |
---|---|

Antipode lattice code | Lattice code constructed via the antipode construction. |

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. |

Body-centered cubic (bcc) lattice code | Three-dimensional lattice consisting of all points \((x,y,z)\) whose integer components are either all even or all odd. |

Coxeter-Todd \(K_{12}\) lattice code | Even integral lattice in dimension \(12\) that exhibits optimal packing. It's automorphism group was discovered by Mitchell [1]. For more details, see [3][2; Sec. 4.9]. |

Dual lattice code | Also called the reciprocal or polar lattice code. For any \(n\)-dimensional lattice \(L\), the dual lattice is \begin{align} L^{\perp} = \{ y\in \mathbb{R}^{n} ~|~ x \cdot y \in \mathbb{Z} ~\forall~ x \in L\}, \tag*{(1)}\end{align} where the Euclidean inner product is used. |

Lattice-based code | Encodes states (codewords) in coordinates of an \(n\)-dimensional lattice, i.e., a discrete set of points in Euclidean space \(\mathbb{R}^n\) that forms a group under vector addition when the set is translated such that one point is at the origin. 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. |

Niemeier lattice code | One of the 24 positive-definite even unimodular lattices of rank 24. |

Root lattice code | A lattice that is symmetric under a specific crystallographic reflection group; see [2; Table 4.1] for the list of crystallographic reflection groups and their corresponding root lattices. The root-lattice family consists of lattices \(A_n\), \(\mathbb{Z}^n\), or \(D_n\) for dimension \(n\), or \(E_{i}\) for \(i\in\{6,7,8\}\). Their generator matrices can be taken to be the root matrices of the corresponding reflection groups. |

Unimodular lattice code | A lattice that is equal to its dual, \(L^\perp = L\). Unimodular lattices have \(\det L = \pm 1\). |

\(A_2\) hexagonal lattice code | Two-dimensional lattice that exhibits optimal packing, solving the packing, kissing, covering and quantization problems. Its dual is the honeycomb lattice. |

\(A_n\) lattice code | Lattice-based \(n\)-dimensional code that can be simply defined in \(n+1\) dimensions as the set of integer vectors \(x\) lying in the hyperplane \(x_0+x_1+\cdots+x_{n} = 0\). |

\(A_n^{\perp}\) lattice code | Stub. |

\(BW_{32}\) Barnes-Wall lattice code | BW lattice in dimension \(32\). |

\(D_3\) face-centered cubic (fcc) lattice code | Laminated three-dimensional lattice consisting of layers of hexagonal lattices. |

\(D_4\) hyper-diamond 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 Construction A. |

\(D_n\) checkerboard lattice code | Lattice code consisting of all points whose coordinates add up to an even integer. |

\(E_6\) root lattice code | Lattice in dimension \(6\). |

\(E_7\) root lattice code | Lattice in dimension \(7\). |

\(E_8\) Gosset lattice code | 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. |

\(\Lambda_{16}\) Barnes-Wall lattice code | BW lattice in dimension \(16\). |

\(\Lambda_{24}\) Leech lattice code | Even unimodular lattice in 24 dimensions that exhibits optimal packing. |

\(\mathbb{Z}^n\) hypercubic lattice code | 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. |

## References

- [1]
- H. H. Mitchell, “Determination of All Primitive Collineation Groups in More than Four Variables which Contain Homologies”, American Journal of Mathematics 36, 1 (1914) DOI
- [2]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
- [3]
- J. H. Conway and N. J. A. Sloane, “The Coxeter–Todd lattice, the Mitchell group, and related sphere packings”, Mathematical Proceedings of the Cambridge Philosophical Society 93, 421 (1983) DOI