Here is a list of codes related to lattices.
Code Description
Antipode lattice Lattice code constructed via the antipode construction.
Barnes-Wall (BW) lattice 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 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 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 For any dimensional lattice \(L\), the dual lattice is the set of vectors whose inner products with the elements of \(L\) are integers.
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 One of the 24 positive-definite even unimodular lattices of rank 24.
Root lattice 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 A lattice that is equal to its dual, \(L^\perp = L\). Unimodular lattices have \(\det L = \pm 1\).
\(A_2\) hexagonal lattice Two-dimensional lattice that exhibits optimal packing, solving the packing, kissing, covering and quantization problems. Its dual is the honeycomb tiling, which is not a lattice (since the points do not form a group under addition) but which consists of two hexagonal lattices. The ruby lattice is a fattened honeycomb tiling interpolating between the honeycomb tiling and hexagonal lattice.
\(A_n\) lattice 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 Lattice-based \(n\)-dimensional code whose codewords form the dual of the \(A_n\) lattice.
\(BW_{32}\) Barnes-Wall lattice BW lattice in dimension \(32\).
\(D_3\) face-centered cubic (fcc) lattice Laminated three-dimensional lattice consisting of layers of hexagonal lattices.
\(D_4\) hyper-diamond lattice 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 Lattice code consisting of all points whose coordinates add up to an even integer.
\(E_6\) root lattice Lattice in dimension \(6\).
\(E_7\) root lattice Lattice in dimension \(7\).
\(E_8\) 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.
\(\Lambda_{16}\) Barnes-Wall lattice BW lattice in dimension \(16\).
\(\Lambda_{24}\) 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.

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