Bounds on the minimum distance of additive quantum codes
Bounds on [[26,15]]2
lower bound: | 4 |
upper bound: | 4 |
Construction
Construction of a [[26,15,4]] quantum code:
[1]: [[40, 30, 4]] quantum code over GF(2^2)
quasicyclic code of length 40 stacked to height 2 with 16 generating polynomials
[2]: [[25, 15, 4]] quantum code over GF(2^2)
Shortening of [1] at { 2, 3, 4, 5, 7, 8, 9, 12, 13, 14, 16, 19, 24, 31, 32 }
[3]: [[26, 15, 4]] quantum code over GF(2^2)
ExtendCode [2] by 1
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 0 1 0 0 0 1 0 0 0|0 1 1 0 0 0 0 1 0 0 0 1 0 1 1 0 1 0 0 1 0 1 0 0 1 0]
[0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 1 0 1 0|1 0 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 1 1 0 1 1 0 1 0 0]
[0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 1 0 0|1 0 0 1 0 0 1 0 0 1 0 1 0 1 0 0 0 1 1 1 1 0 0 1 1 0]
[0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 1 0 0|1 1 1 0 1 1 0 1 1 1 1 0 0 1 0 1 0 0 1 0 1 1 0 0 1 0]
[0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 1 1 0 0 1 0 1 0|0 0 1 0 0 0 1 1 0 0 1 1 1 1 1 1 0 1 0 1 0 1 1 1 0 0]
[0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 0 0 1 0 1 0 0 0|1 1 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 0 0 1 0]
[0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 1 0 1 1 0 0 1 0 1 0 0|0 1 0 0 0 1 1 0 0 1 0 0 1 1 1 0 1 0 1 1 1 1 1 0 0 0]
[0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 0 1 0 1 1 1 0 0 1 0 0|1 1 1 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0]
[0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 0 1 1 0 1 0 1 0 1 0 0|1 1 0 1 0 0 0 0 1 0 1 0 0 0 1 0 1 1 0 1 1 0 0 0 1 0]
[0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 1 1 0 1 1 1 1 0|0 1 1 1 1 0 0 0 1 1 1 0 1 0 1 1 1 1 1 1 0 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|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]
last modified: 2005-06-24
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