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Unimodular lattice

Alternative names: Self-dual lattice.

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].

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 code over \(R\)— There are parallels between self-dual codes over \(\mathbb{Z}_{2k}\) and even unimodular lattices [4,5].
  • Barnes-Wall (BW) lattice— Unions of certain RM codes yield self-dual quaternary codes over \(\mathbb{Z}_4\) that then give rise to certain BW lattices [6,7].
  • Spherical design— A union of \(t\) shells of self-dual lattices and their shadows form spherical \(t\)-designs [8].

Primary Hierarchy

Parents
Unimodular lattice
Children
Niemeier lattices are even and unimodular.
The \(E_8\) Gosset lattice is even and unimodular.
The hypercubic lattice is odd and unimodular.

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]
A. Bonnecaze, P. Sole, C. Bachoc, and B. Mourrain, “Type II codes over Z/sub 4/”, IEEE Transactions on Information Theory 43, 969 (1997) DOI
[5]
E. Bannai, S. T. Dougherty, M. Harada, and M. Oura, “Type II codes, even unimodular lattices, and invariant rings”, IEEE Transactions on Information Theory 45, 1194 (1999) DOI
[6]
P. Sole, "Generalized theta functions for lattice vector quantization", in Coding and Quantization, DIMACS Series in Dr,crete Mathenulies and Theoretical Computer Science, vol. 14. Providence, RH: American Math. Soc., 1993, pp. 27-32.
[7]
A. Bonnecaze, P. Sole, and A. R. Calderbank, “Quaternary quadratic residue codes and unimodular lattices”, IEEE Transactions on Information Theory 41, 366 (1995) DOI
[8]
C. Pache, “Shells of selfdual lattices viewed as spherical designs”, (2005) arXiv:math/0502313
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Zoo Code ID: self_dual_lattice

Cite as:
“Unimodular lattice”, 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}, 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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“Unimodular lattice”, 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.