# Body-centered cubic (bcc) lattice code

## Description

Three-dimensional lattice consisting of all points \((x,y,z)\) whose integer components are either all even or all odd.

## Protection

The bcc lattice has density \(\Delta=\pi\sqrt{3}/8\approx 0.6802\). It exhibits the thinnest lattice covering [1] in three dimensions. It solves the lattice quantizer problem in three dimensions with \(G_3 = \frac{19}{192\cdot 2^{1/3}}\approx 0.0785\) [2].

## Parent

- \(A_n^{\perp}\) lattice code — The bcc lattice is the dual of the \(A_3=D_3\) fcc lattice.

## Cousins

- \(D_3\) face-centered cubic (fcc) lattice code — The bcc and fcc lattices are dual to each other.
- Dual lattice code — The bcc and fcc lattices are dual to each other.
- Raussendorf-Bravyi-Harrington (RBH) cluster-state code — The RBH code is defined on the bcc lattice.

## References

- [1]
- Bambah, R. P., and H. Gupta. "On lattice coverings by spheres." Proceedings of the National Institute of Sciences of India. Vol. 20. Indian National Science Academy, 1954.
- [2]
- E. S. Barnes and N. J. A. Sloane, “The Optimal Lattice Quantizer in Three Dimensions”, SIAM Journal on Algebraic Discrete Methods 4, 30 (1983) DOI

## Page edit log

- Victor V. Albert (2022-11-26) — most recent

## Cite as:

“Body-centered cubic (bcc) lattice code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/bcc