Gnu code

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


 * $$|0_{\rm L}\rangle = \sum_{\ell\, \textrm{even}\atop 0\le\ell\le n} \sqrt{\frac{2^{n-1}}} |D^m_{g\ell}\rangle$$
 * $$|1_{\rm L}\rangle = \sum_{\ell\, \textrm{odd}\atop 0\le\ell\le n} \sqrt{\frac{2^{n-1}}} |D^m_{g\ell}\rangle$$

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


 * $$|D^4_2\rangle = \frac{|0011\rangle + |0101\rangle + |1001\rangle + |0110\rangle + |1010\rangle + |1100\rangle}{\sqrt{6}}$$

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