\(A_n\) lattice code 

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 determinant \(n+1\), 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 density \(V_n/2^n\) (with \(V_n\) the volume of the unit \(n\)-sphere).

Parent

Cousin

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Zoo Code ID: an

Cite as:
\(A_n\) lattice code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/an
BibTeX:
@incollection{eczoo_an, title={\(A_n\) lattice code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/an} }
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Permanent link:
https://errorcorrectionzoo.org/c/an

Cite as:

\(A_n\) lattice code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/an

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/analog/lattice/root/an.yml.