## Bounds on the minimum distance of additive quantum codes

### Bounds on [[16,6]]_{3}

lower bound: | 5 |

upper bound: | 5 |

### Construction

Construction type: BagFawzi

Construction of a [[16,6,5]] quantum code:
[1]: [[16, 6, 5]] quantum code over GF(3^2)
cyclic code of length 16 with generating polynomial x^12 + x^11 + 2*x^10 + w^2*x^9 + w^6*x^8 + w^3*x^7 + w^7*x^6 + 2*x^5 + x^4 + w^3*x^3 + w^2*x^2 + 1
stabilizer matrix:
[1 0 0 0 0 0 1 2 0 2 0 2 2 0 2 1|0 0 0 0 0 2 0 1 2 0 0 2 2 0 1 2]
[0 0 0 0 0 1 0 0 2 1 0 2 2 0 1 0|1 0 0 0 0 0 1 1 1 1 0 1 0 0 1 2]
[0 1 0 0 0 1 2 2 1 2 2 0 2 2 2 1|0 0 0 0 0 1 0 0 0 0 0 2 0 2 0 1]
[0 0 0 0 0 1 1 0 2 0 1 2 1 2 1 1|0 1 0 0 0 0 1 2 2 2 1 1 1 0 1 0]
[0 0 1 0 0 2 0 0 0 1 2 1 2 2 2 1|0 0 0 0 0 1 0 1 0 2 0 0 0 0 0 2]
[0 0 0 0 0 0 0 2 0 0 0 2 0 1 0 0|0 0 1 0 0 1 0 0 0 2 2 2 2 1 2 2]
[0 0 0 1 0 1 1 1 2 2 1 2 1 2 1 1|0 0 0 0 0 1 2 0 0 1 2 2 1 0 0 0]
[0 0 0 0 0 1 0 0 1 1 0 2 1 0 2 0|0 0 0 1 0 0 2 1 1 1 2 0 2 2 2 1]
[0 0 0 0 1 0 0 2 1 0 2 2 0 1 0 0|0 0 0 0 0 1 1 1 1 0 1 0 0 1 2 1]
[0 0 0 0 0 2 1 0 1 0 1 1 0 1 2 2|0 0 0 0 1 0 2 1 0 0 1 1 0 2 1 0]
last modified: 2024-06-10

### 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 necessarily 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
(codes@codetables.de).
Last change: 10.06.2024