## Bounds on the minimum distance of additive quantum codes

### Bounds on [[19,11]]2

 lower bound: 3 upper bound: 3

### Construction

```Construction of a [[19,11,3]] quantum code:
[1]:  [[17, 11, 3]] Quantum code over GF(2^2)
Construction from a stored generator matrix
[2]:  [[19, 11, 3]] Quantum code over GF(2^2)
ExtendCode [1] by 2

stabilizer matrix:

[1 0 0 0 0 1 0 1 1 0 1 0 1 1 0 0 0 0 0|1 1 1 0 1 0 0 1 0 0 1 1 1 0 0 1 1 0 0]
[0 1 0 0 0 1 0 1 1 0 1 0 1 1 0 1 1 0 0|1 0 1 0 1 1 1 0 1 0 0 0 0 1 0 1 0 0 0]
[0 0 1 0 0 1 0 1 1 1 0 1 0 0 1 1 1 0 0|0 0 0 0 1 1 1 1 0 0 1 1 0 0 0 0 1 0 0]
[0 0 0 1 0 0 0 1 1 1 1 1 0 1 1 1 1 0 0|0 0 1 0 0 1 0 1 1 1 0 0 1 1 1 0 0 0 0]
[0 0 0 0 1 1 0 0 1 0 0 1 0 1 0 1 1 0 0|1 1 1 1 0 0 0 1 1 1 0 1 0 1 0 0 1 0 0]
[0 0 0 0 0 0 1 1 1 1 0 1 0 1 0 0 1 0 0|0 1 0 1 1 0 1 0 1 0 1 1 1 0 1 1 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 1|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

```

### 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.