Description
Lattice-based \(n\)-dimensional code that can be simply defined in \(n+1\) dimensions as the set of integer vectors \(x\) lying in the hyperplane \(x_0+x_1+\cdots+x_{n} = 0\).
Its generator matrix can be represented by \begin{align} G=\begin{pmatrix}-1 & \phantom{-}1 & 0 & 0 & \cdots & 0 & 0\\ 0 & -1 & \phantom{-}1 & 0 & \cdots & 0 & 0\\ 0 & 0 & -1 & \phantom{-}1 & \cdots & 0 & 0\\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ 0 & 0 & 0 & 0 & \cdots & -1 & \phantom{-}1 \end{pmatrix}~. \tag*{(1)}\end{align}
Protection
The lattice has Gram determinant \(n+1\) (equivalently, fundamental volume \(\sqrt{n+1}\)), kissing number \(n(n+1)\), packing radius \(1/\sqrt{2}\), covering radius \(\sqrt{\frac{a\left(n+1-a\right)}{n+1}}\) (with \(a=\lfloor (n+1)/2 \rfloor\)), and center density \(V_n/(2^{n/2}\sqrt{n+1})\) (with \(V_n\) the volume of the unit \(n\)-sphere) [1; Chs. 4 and 21].Cousin
Member of code lists
Primary Hierarchy
Parents
\(A_n\) lattice
Children
The \(D_3\) fcc lattice is equivalent to the \(A_3\) root lattice [1; Ch. 1, pg. 13].
References
- [1]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
Page edit log
- Victor V. Albert (2023-05-15) — most recent
Cite as:
“\(A_n\) lattice”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/an