## Bounds on the minimum distance of additive quantum codes

### Bounds on [[20,12]]_{2}

lower bound: | 3 |

upper bound: | 3 |

### Construction

Construction of a [[20,12,3]] quantum code:
[1]: [[40, 33, 3]] Quantum code over GF(2^2)
Construction from a stored generator matrix
[2]: [[19, 12, 3]] Quantum code over GF(2^2)
Shortening of [1] at { 3, 4, 8, 9, 14, 15, 16, 17, 18, 21, 24, 27, 29, 31, 32, 33, 35, 36, 38, 39, 40 }
[3]: [[20, 12, 3]] Quantum code over GF(2^2)
ExtendCode [2] by 1
stabilizer matrix:
[1 0 0 1 0 0 1 0 1 1 0 0 1 1 0 0 0 0 1 0|1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 1 0 1 0 0]
[0 1 0 1 0 0 0 1 1 1 0 0 0 1 1 0 0 0 1 0|1 0 1 0 0 0 1 1 0 1 1 1 0 0 1 0 0 1 1 0]
[0 0 1 1 0 0 1 1 0 0 0 0 1 1 0 1 1 1 1 0|0 1 1 0 1 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0]
[0 0 0 0 1 0 1 0 1 0 1 0 1 0 0 0 1 1 0 0|0 0 1 1 1 0 0 0 0 0 0 1 1 0 1 1 0 1 1 0]
[0 0 0 0 0 1 1 1 1 1 1 0 0 0 1 1 0 1 1 0|1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 0 0 0|1 1 1 1 1 0 0 0 0 1 1 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 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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0]
last modified: 2005-06-29

### 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