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 \(\det L = \pm 1\).

Protection

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].

Parent

Children

Cousins

  • 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].

References

[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
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Zoo Code ID: self_dual_lattice

Cite as:
“Unimodular lattice code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/self_dual_lattice
BibTeX:
@incollection{eczoo_self_dual_lattice, title={Unimodular lattice code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/self_dual_lattice} }
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Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/analog/lattice/dual/self_dual_lattice.yml.