# 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

- Dual linear code — Dual lattices are lattice analogues of dual codes. There are several parallels between (doubly-even) self-dual binary codes and (even) unimodular lattices [2][3]. There are also parallels between self-dual codes over \(\mathbb{Z}_{2k}\) and even unimodular lattices [4].
- \(\Lambda_{24}\) Leech lattice code — The \(\Lambda_{24}\) Leech lattice code is even and unimodular.
- Body-centered cubic (bcc) lattice code — The bcc and fcc lattices are dual to each other.
- Niemeier lattice code — The Niemeier lattice codes is even and unimodular.
- \(D_4\) hyper-diamond lattice code — The \(D_4\) hyper-diamond lattice is self-dual.
- \(E_8\) Gosset lattice code — The \(E_8\) Gosset lattice code is even and unimodular.
- \(\mathbb{Z}^n\) hypercubic lattice code — The hypercubic lattice code 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]
- E. Bannai et al., “Type II codes, even unimodular lattices, and invariant rings”, IEEE Transactions on Information Theory 45, 1194 (1999) DOI

## Page edit log

- Victor V. Albert (2022-02-25) — most recent

## Cite as:

“Dual lattice code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/dual_lattice