Unimodular lattice code

Description

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

The minimum norm of a unimodular lattice satisfies \begin{align} \mu\leq2\left[\frac{n}{24}\right]+2~, \tag*{(1)}\end{align} unless \(n = 23\) [1].

Niemeier lattice code — Niemeier lattice codes are even and unimodular.
\(D_4\) hyper-diamond lattice code
\(E_8\) Gosset lattice code — The \(E_8\) Gosset lattice code is even and unimodular.
\(\mathbb{Z}^n\) hypercubic lattice code — The hypercubic lattice code is odd and unimodular.

Self-dual linear code — Unimodular lattices are lattice analogues of self-dual codes. There are several parallels between (doubly-even) self-dual binary codes and (even) unimodular lattices [2,3].
Self-dual additive code — There are parallels between self-dual additive codes over \(\mathbb{Z}_{2k}\) and even unimodular lattices [4].

[1]: E. M. Rains and N. J. A. Sloane, “The Shadow Theory of Modular and Unimodular Lattices”, Journal of Number Theory 73, 359 (1998) DOI
[2]: J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
[3]: Self-Dual Codes and Invariant Theory (Springer-Verlag, 2006) DOI
[4]: E. Bannai et al., “Type II codes, even unimodular lattices, and invariant rings”, IEEE Transactions on Information Theory 45, 1194 (1999) DOI

Page edit log

“Unimodular lattice code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/self_dual_lattice