## Bounds on the minimum distance of additive quantum codes

### Bounds on [[22,13]]_{2}

lower bound: | 3 |

upper bound: | 4 |

### Construction

Construction of a [[22,13,3]] quantum code:
[1]: [[21, 15, 3]] Quantum code over GF(2^2)
Construction from a stored generator matrix
[2]: [[21, 13, 3]] Quantum code over GF(2^2)
Subcode of [1]
[3]: [[22, 13, 3]] Quantum code over GF(2^2)
ExtendCode [2] by 1
stabilizer matrix:
[1 0 0 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 0 1 0|0 1 1 0 1 0 1 1 0 1 0 0 1 0 1 1 1 1 0 0 0 0]
[0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 0 1 0 1 1 0|1 0 1 1 1 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 0]
[0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0|0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0]
[0 0 0 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 1 0|1 1 0 0 0 0 0 1 1 0 0 1 1 0 0 1 0 0 0 1 0 0]
[0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 0|0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0]
[0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 1 0 0|0 1 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0|0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0|0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0]
last modified: 2005-06-30

### Notes

- All codes establishing the lower bounds where constructed using MAGMA.
- Most upper bounds on qubit codes for
*n≤100* are based on a MAGMA program by Eric Rains.
- For
*n>100*, the upper bounds on qubit codes are weak (and not even monotone in *k*).
- Some additional information can be found in the book by Nebe, Rains, and Sloane.
- My apologies to all authors that have contributed codes to this table for not giving specific credits.

This page is maintained by
Markus Grassl
(grassl@ira.uka.de).
Last change: 23.10.2014