## Bounds on the minimum distance of additive quantum codes

### Bounds on [[13,5]]_{2}

lower bound: | 3 |

upper bound: | 3 |

### Construction

Construction of a [[13,5,3]] quantum code:
[1]: [[11, 5, 3]] Quantum code over GF(2^2)
Construction from a stored generator matrix
[2]: [[13, 5, 3]] Quantum code over GF(2^2)
ExtendCode [1] by 2
stabilizer matrix:
[1 0 0 0 1 0 1 1 0 1 0 0 0|1 0 1 0 1 1 1 1 1 0 0 0 0]
[0 1 0 0 0 0 1 0 1 1 0 0 0|1 0 0 0 1 1 0 0 1 0 1 0 0]
[0 0 1 0 0 0 1 1 0 0 1 0 0|1 0 0 1 0 1 0 1 0 1 0 0 0]
[0 0 0 1 1 0 0 1 1 1 1 0 0|0 0 0 0 0 1 1 1 1 1 1 0 0]
[0 0 0 0 0 1 1 1 1 1 1 0 0|0 0 0 1 1 1 1 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 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 1 1 1 1 1 1 1 1 1 1 0 0]
last modified: 2011-06-23

### Further notes

The upper bound was shown in

Jürgen Bierbrauer, Richard Fears, Stefano Marcugini, and Fernanda Pambianco,

"The Nonexistence of a [[13,5,4]]-Quantum Stabilizer Code,"

*IEEE Transactions on Information Theory*, 57(7):4788-4793 (2011).

DOI: 10.1109/TIT.2011.2146430

### 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