In quantum information, the gnu code refers to a particular family of quantum error correcting codes, with the special property of being invariant under permutations of the qubits. Given integers g (the gap), n (the occupancy), and m (the length of the code), the two codewords are

${\displaystyle |0_{\rm {L))\rangle =\sum _{\ell \,{\textrm {even)) \atop 0\leq \ell \leq n}{\sqrt {\frac {n \choose \ell }{2^{n-1))))|D_{g\ell }^{m}\rangle }$
${\displaystyle |1_{\rm {L))\rangle =\sum _{\ell \,{\textrm {odd)) \atop 0\leq \ell \leq n}{\sqrt {\frac {n \choose \ell }{2^{n-1))))|D_{g\ell }^{m}\rangle }$

where ${\displaystyle |D_{k}^{m}\rangle }$ are the Dicke states consisting of a uniform superposition of all weight-k words on m qubits, e.g.

${\displaystyle |D_{2}^{4}\rangle ={\frac {|0011\rangle +|0101\rangle +|1001\rangle +|0110\rangle +|1010\rangle +|1100\rangle }{\sqrt {6))))$

The real parameter ${\displaystyle u={\frac {m}{gn))}$ scales the density of the code. The length ${\displaystyle m=gnu}$, hence the name of the code. For odd ${\displaystyle g=n}$ and ${\displaystyle u\geq 1}$, the gnu code is capable of correcting ${\displaystyle {\frac {g-1}{2))}$ erasure errors,[1] or deletion errors.[2]

## References

1. ^ Ouyang, Yingkai (2014-12-10). "Permutation-invariant quantum codes". Physical Review A. 90 (6): 062317. arXiv:1302.3247. Bibcode:2014PhRvA..90f2317O. doi:10.1103/physreva.90.062317. ISSN 1050-2947. S2CID 119114455.
2. ^ Ouyang, Yingkai (2021-02-04). "Permutation-invariant quantum coding for quantum deletion channels". arXiv:2102.02494v1 [quant-ph].