Description
A lattice, scaled to be integral, 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}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 [1,3,4]. Even self-dual binary codes and even unimodular lattices define CFTs [5–7].
- Self-dual code over \(\mathbb{Z}_q\)— There are parallels between self-dual codes over \(\mathbb{Z}_{2k}\) and even unimodular lattices [8,9]. Type I (type II) codes over \(\mathbb{Z}_4\) yield type I (type II) lattices under Construction \(A_4\) [10; Thm. 12.5.12].
- Construction \(A_4\) lattice— Type I (type II) codes over \(\mathbb{Z}_4\) yield type I (type II) lattices under Construction \(A_4\) [10; Thm. 12.5.12].
- Conformal-field theory (CFT) code— Even self-dual binary codes and even unimodular lattices define CFTs [5–7].
- Coxeter-Todd \(K_{12}\) lattice— The Coxeter-Todd lattice is unimodular when defined as a complex lattice over the Eisenstein integers [1; Ch. 4, pg. 128].
- Self-dual polytope code— Self-dual polytope codes are spherical analogues of self-dual lattices.
- Spherical design— A union of \(t\) shells of self-dual lattices and their shadows form spherical \(t\)-designs [11].
Member of code lists
Primary Hierarchy
References
- [1]
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
- [2]
E. M. Rains and N. J. A. Sloane, “The Shadow Theory of Modular and Unimodular Lattices”, Journal of Number Theory 73, 359 (1998) DOI
- [3]
Self-Dual Codes and Invariant Theory (Springer-Verlag, 2006) DOI
- [4]
J. Henriksson and B. McPeak, “Averaging over codes and an \(SU(2)\) modular bootstrap”, (2023) arXiv:2208.14457
- [5]
K. S. NARAIN, “New Heterotic String Theories in Uncompactified Dimensions < 10”, Current Physics–Sources and Comments 246 (1989) DOI
- [6]
L. Dolan, P. Goddard, and P. Montague, “Conformal field theory, triality and the monster group”, Physics Letters B 236, 165 (1990) DOI
- [7]
L. Dolan, P. Goddard, and P. Montague, “Conformal field theories, representations and lattice constructions”, Communications in Mathematical Physics 179, 61 (1996) arXiv:hep-th/9410029 DOI
- [8]
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
- [9]
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
- [10]
W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
- [11]
C. Pache, “Shells of selfdual lattices viewed as spherical designs”, (2005) arXiv:math/0502313
Page edit log
- Victor V. Albert (2026-06-08) — most recent
- Victor V. Albert (2023-04-22)
Cite as:
“Unimodular lattice”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/self_dual_lattice, arXiv:2606.11484