Bounds on the minimum distance of additive quantum codes
Bounds on [[30,7]]2
lower bound: | 7 |
upper bound: | 9 |
Construction
Construction of a [[30,7,7]] quantum code:
[1]: [[30, 8, 7]] quantum code over GF(2^2)
cyclic code of length 30 with generating polynomial x^29 + x^28 + w*x^27 + x^26 + w^2*x^24 + x^23 + w*x^22 + x^20 + x^18 + w^2*x^17 + w*x^16 + w*x^15 + x^14 + w^2*x^13 + 1
[2]: [[30, 7, 7]] quantum code over GF(2^2)
Subcode of [1]
stabilizer matrix:
[1 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 1 0|1 0 1 1 0 1 0 1 0 1 1 1 1 0 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 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 1|0 1 0 1 1 0 1 0 1 1 1 1 1 1 0 1 1 1 0 1 1 0 1 0 0 0 1 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 1 0 1 0 0 0 1 0|0 0 0 1 1 0 0 0 0 0 1 0 0 1 0 1 0 1 1 0 1 0 0 1 0 1 0 1 0 1]
[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 1 0 0 1 1|1 0 0 0 1 1 0 0 0 1 0 1 0 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1]
[0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 0|1 1 1 1 0 0 1 1 0 1 1 1 0 0 1 0 1 1 0 1 0 0 0 1 1 1 1 0 1 1]
[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 1 0 1 1 1|1 1 1 1 1 0 0 1 1 1 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 0 0 0 1 0]
[0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0|0 1 0 0 1 0 0 1 1 0 1 0 0 1 1 1 0 0 1 1 1 1 1 1 1 1 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 1 0 1 1 0|0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1]
[0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0|0 0 0 1 0 0 1 0 0 1 1 0 1 0 0 1 1 1 0 0 1 1 1 1 1 1 1 1 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 1 0 1 0 1|0 0 0 0 1 0 0 1 0 0 1 1 0 1 0 0 1 1 1 0 0 1 1 1 1 1 1 1 1 0]
[0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0|1 0 1 1 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 0 1]
[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 1 1 0|1 1 0 1 1 0 0 0 1 0 1 1 0 0 0 0 1 1 1 0 0 0 1 1 1 1 0 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0|0 1 1 0 1 1 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 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 1 0 0 0 1|0 0 1 1 0 1 1 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0|1 0 1 0 1 1 1 0 0 1 1 0 0 1 0 1 1 0 0 1 0 1 0 0 0 0 1 1 1 1]
[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 1 1 1|1 1 0 1 0 1 1 1 0 1 1 1 0 0 1 0 1 1 0 1 0 1 0 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 0 0 0 1 0 1 0 0 0 0 0|0 1 0 1 1 1 1 0 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 0 1 0 1 1 0 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 1 0 0 1 0|1 0 1 0 1 1 1 1 0 0 1 1 0 0 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0|0 1 0 1 0 1 1 1 1 0 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 0 1 0 1 1]
[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 1 0 1 0 0|1 0 1 0 1 0 1 1 1 1 0 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 0 1 0 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 1 0 1 0|1 1 0 1 0 1 0 1 1 1 1 0 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 0 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 1 0 1|0 1 1 0 1 0 1 0 1 1 1 1 0 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 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 1 0 0 0 0 1 0|0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1]
last modified: 2005-06-27
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
(codes@codetables.de).
Last change: 23.10.2014