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 \(\mathbb{Z}_q\)— There are parallels between self-dual codes over \(\mathbb{Z}_{2k}\) and even unimodular lattices [4,5]. Type I (type II) codes over \(\mathbb{Z}_4\) yield type I (type II) lattices under Construction \(A_4\) [6; Thm. 12.5.12].
- Construction \(A_4\) code— Type I (type II) codes over \(\mathbb{Z}_4\) yield type I (type II) lattices under Construction \(A_4\) [6; Thm. 12.5.12].
- 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 [7,8].
- Spherical design— A union of \(t\) shells of self-dual lattices and their shadows form spherical \(t\)-designs [9].
Member of code lists
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]
- W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
- [7]
- 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.
- [8]
- A. Bonnecaze, P. Sole, and A. R. Calderbank, “Quaternary quadratic residue codes and unimodular lattices”, IEEE Transactions on Information Theory 41, 366 (1995) DOI
- [9]
- C. Pache, “Shells of selfdual lattices viewed as spherical designs”, (2005) arXiv:math/0502313
Page edit log
- Victor V. Albert (2023-04-22) — most recent
Cite as:
“Unimodular lattice”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/self_dual_lattice