Dual lattice code

Description

Also called the reciprocal or polar lattice code. For any \(n\)-dimensional lattice \(L\), the dual lattice is \begin{align} L^{\perp} = \{ y\in \mathbb{R}^{n} ~|~ x \cdot y \in \mathbb{Z} ~\forall~ x \in L\}, \tag*{(1)}\end{align} where the Euclidean inner product is used.

A lattice that is contained in its dual, \(L \subseteq L^\perp\), is called integral. The Gram matrix of such a lattice has integer entries, and its dual is contained in a suitably scaled version of itself, \(L^{\perp} \subseteq L/\det L\). Integral lattices are classified into even or odd, where the norm squared of every lattice vector is an even or odd integer, respectively.

A lattice that is equal to its dual, \(L^\perp = L\), is called unimodular or self-dual. Unimodular lattices have \(\det L = \pm 1\).

Protection

The Gram matrix of \(L^{\perp}\) is the inverse of that of \(L\). The generator matrix of \(L^{\perp}\) is the transposed inverse of that of \(L\).

The minimum norm of a unimodular lattice satisfies \begin{align} \mu\leq2\left[\frac{n}{24}\right]+2~, \tag*{(2)}\end{align} unless \(n = 23\) [1].

Parent

Cousins

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: dual_lattice

Cite as:
“Dual lattice code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/dual_lattice
BibTeX:
@incollection{eczoo_dual_lattice, title={Dual lattice code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/dual_lattice} }
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“Dual lattice code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/dual_lattice

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