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 }](https://wikimedia.org/api/rest_v1/media/math/render/svg/149d75d5690d4c1a4ca4897e03dc6e2a321c27fb)
![{\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 }](https://wikimedia.org/api/rest_v1/media/math/render/svg/09c2f7398d1aff307544f536018221d0c42779c9)
where
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))))](https://wikimedia.org/api/rest_v1/media/math/render/svg/a212c4fe3c1076a783b04f11d0cea9bd19b6c58d)
The real parameter
scales the density of the code. The length
, hence the name of the code. For odd
and
, the gnu code is capable of correcting
erasure errors,[1] or deletion errors.[2]