Here is a list of codes related to lattices.

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Code Description
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.
Construction \(A_4\) code A lattice that is constructed from a linear code over \(\mathbb{Z}_4\) using Construction \(A_4\).
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 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 \(\det L = \pm 1\).
\(A_2\) triangular lattice Two-dimensional lattice that corresponds to the triangular tiling and that exhibits optimal packing, solving the packing, kissing, covering and quantization problems. As a tiling, its dual (whose points lie at the centers of each triangle) is the honeycomb tiling.
\(A_n\) 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 triangular lattices.
\(D_4\) hyper-diamond lattice BW lattice in dimension \(4\), which is the lattice corresponding to the \([4,3,2]\) SPC codes via Construction A. The lattice points form the \(\{3,3,4,3\}\) tesselation of 4-space [4; pg. 136].
\(D_n\) checkerboard lattice Lattice 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
[4]
H. S. M. Coxeter. Regular polytopes. Courier Corporation, 1973.