Unimodular lattice code

Also known as Self-dual lattice code.

Description

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].
Spherical design — A union of \(t\) shells of self-dual lattices and their shadows form spherical \(t\)-designs [5].

[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
[5]: C. Pache, “Shells of selfdual lattices viewed as spherical designs”, (2005) arXiv:math/0502313

Page edit log

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