Description
Lattice code consisting of all points whose coordinates add up to an even integer.
Its generator matrix can be represented by \begin{align} G=\begin{pmatrix}-1 & -1 & 0 & 0 & \cdots & 0 & 0\\ \phantom{-}1 & -1 & 0 & 0 & \cdots & 0 & 0\\ 0 & \phantom{-}1 & -1 & 0 & \cdots & 0 & 0\\ 0 & 0 & \phantom{-}1 & -1 & \cdots & 0 & 0\\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ 0 & 0 & 0 & 0 & \cdots\phantom{-} & 1 & -1 \end{pmatrix}~. \tag*{(1)}\end{align}
Protection
Exhibits the densest lattice packing and highest known kissing number in four and five [1] dimensions.
Parents
- Root lattice code
- Construction-\(A\) code — \([n,n-1,2]\) SPC codes yield \(D_n\) checkerboard lattice codes via Construction A [2; Ex. 10.5.1].
Children
- \(D_4\) hyper-diamond lattice code
- \(D_3\) face-centered cubic (fcc) lattice code — The \(D_3\) root lattice is equivalent to the \(A_3\) root lattice [2; Ch. 10].
Cousin
- Single parity-check (SPC) code — \([n,n-1,2]\) SPC codes yield \(D_n\) checkerboard lattice codes via Construction A [2; Ex. 10.5.1].
References
- [1]
- Sur les formes quadratiques positives. (Zus. Mit S. Zolotareff) Korkine in: Mathematische Annalen
- [2]
- T. Ericson, and V. Zinoviev, eds. Codes on Euclidean spheres. Elsevier, 2001.
Page edit log
- Victor V. Albert (2022-12-12) — most recent
Cite as:
“\(D_n\) checkerboard lattice code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/dn