Bounds on the minimum distance of additive quantum codes
Bounds on [[88,68]]2
lower bound: | 5 |
upper bound: | 6 |
Construction
Construction of a [[88,68,5]] quantum code:
[1]: [[122, 104, 5]] quantum code over GF(2^2)
Construction from a stored generator matrix
[2]: [[86, 68, 5]] quantum code over GF(2^2)
Shortening of [1] at { 49, 62, 69, 70, 72, 73, 74, 76, 77, 79, 80, 81, 83, 90, 91, 92, 93, 94, 97, 98, 99, 100, 101, 103, 107, 108, 109, 110, 111, 112, 113, 115, 116, 118, 119, 121 }
[3]: [[88, 68, 5]] quantum code over GF(2^2)
ExtendCode [2] by 2
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 1 1 1 1 1 1 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 1 0 1 1 1 1 0 1 0 0 0 1 1 0 1 1 1 1 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0|1 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 0 1 0 0 1 0 0 1 0 1 0 1 0 0 0 1 0 1 0 1 1 0 0 0 1 1 0 1 1 0 1 1 0 0 0 1 0 0 1 0 1 1 0 1 1 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 0 0 1 1 0 1 1 1 0 0 0 1 0 1 0 0 0 1 1 0 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 0 0 0 0 0 1 0 1 0 1 0 1 0 0|0 0 1 0 1 1 0 0 1 1 1 0 1 0 1 1 0 1 0 1 0 0 1 1 1 1 1 1 0 0 1 0 0 0 1 1 1 1 1 0 0 1 1 1 0 0 0 0 1 0 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 1 0 0 1 1 1 1 1 1 1 1 0 0 0 1 1 1 0 1 1 1 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 1 0 1 1 1 0 0 0 1 1 1 1 0 1 0 0 1 0 1 1 1 1 1 0 0 1 0 1 0 0 0 1 0 1 1 0 0 0 1 0 0 0 0 1 1 0 1 1 0 0 0 0 0 1 0 1 0 0 1 0 0|1 1 0 0 0 1 1 0 1 1 1 0 0 0 1 0 1 1 0 1 1 0 0 1 1 1 0 0 1 1 0 0 1 0 0 1 0 1 0 0 0 1 1 1 1 0 1 1 0 1 1 0 1 1 0 1 0 0 0 1 0 1 1 1 0 1 1 0 1 1 1 1 0 0 0 1 1 0 1 1 0 1 0 0 1 0 0 0]
[0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 0 0 0 0 1 1 0 1 0 0 0 0 1 1 1 1 0 1 0 1 0 1 1 1 0 1 1 0 0|1 1 0 1 1 1 1 1 0 0 1 1 1 0 0 0 1 0 1 0 1 0 1 0 1 0 0 1 1 1 1 0 0 1 0 0 0 1 0 0 0 0 0 1 1 1 1 0 1 1 0 1 1 1 1 0 1 1 0 0 0 0 1 1 0 0 1 1 0 1 0 1 1 0 0 1 0 0 0 1 1 0 0 0 1 1 0 0]
[0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 1 0 1 0 1 0 1 1 0 0 1 1 1 1 1 0 1 1 1 0 0 0 1 0 1 1 0 0 1 0 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 0 1 0 0 0 0 0|1 0 1 0 0 1 0 1 1 0 1 1 0 0 1 1 0 1 1 0 1 0 1 0 1 0 1 1 1 1 0 0 0 1 0 0 1 1 0 1 1 0 0 1 1 1 0 0 1 1 0 1 0 1 1 1 1 0 1 0 1 0 1 0 0 1 0 1 1 0 1 1 1 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0]
[0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 1 1 0 0 0 1 0 0 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 1 1 1 1 0 0 0 0 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 0 0 0 1 0 1 0 1 1 1 1 0 0|0 0 0 0 1 0 0 1 0 1 1 1 0 0 1 0 1 1 1 0 0 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 0 0 1 0 0 0 0 0 1 1 1 1 1 1 1 0 1 0 0 1 1 1 1 0 1 1 0 0 0 0 0 0 1 0 0 0 1 1 0 1 0 1 0 1 1 0 0 1 0 0 0]
[0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 1 1 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 0 0 1 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0|0 0 0 1 0 0 1 1 0 1 1 0 1 0 0 1 0 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 0 0]
[0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 1 0 1 0 0 1 1 1 1 1 0 0 0 0 1 1 0 1 0 0 0 0 1 1 0 1 1 1 0 0 0 0 1 0 0 0 1 0 1 1 0 0|1 1 0 0 0 0 1 0 0 1 1 1 0 0 0 0 1 0 1 1 1 0 0 0 0 0 1 0 1 1 0 0 1 1 1 0 1 1 1 0 1 1 0 0 0 1 0 1 0 1 1 1 1 1 1 1 0 0 1 1 1 0 1 1 1 0 0 1 0 0 0 0 1 0 0 1 1 1 1 0 1 0 0 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 0 1 0 0 0 1 1 1 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 0 1 1 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0 1 1 0 0 1 1 0 1 1 0 1 0 1 0 1 1 0 0 1 0 0 0|1 1 1 1 1 0 1 0 0 1 0 1 0 0 1 1 0 0 1 1 1 0 1 0 0 0 1 1 1 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 0 1 0 0 0 0 0 0 0 1 0 1 0 0 1 0 1 1 1 0 0 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0]
[0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 0 0 1 0 1 1 1 0 1 1 0 1 1 0 0 1 1 0 1 1 0 1 1 0 0 0 1 1 0 0 0 1 1 0 1 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 0 0 1 0 1 0 0 0|1 1 0 1 1 0 1 0 0 0 0 1 1 0 1 1 0 0 1 1 1 1 0 1 0 1 0 0 1 0 1 0 0 1 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 1 1 1 0 0 1 1 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 0 1 0 1 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 1 0 0 0 1 0 1 1 1 0 0 0 1 0 1 1 0 0 1 0 1 1 1 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 0 0 0 1 0 1 1 1 1 1 0 1 1 1 0 0|0 0 1 1 0 1 1 0 1 0 0 0 0 1 1 0 1 1 0 0 1 1 1 0 0 1 0 1 0 1 0 0 1 0 0 1 1 0 1 1 1 0 1 0 1 1 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 0 1 1 0 1 1 0 1 0 0 1 0 0 1 1 0 1 1 1 0 0 0 0 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 1 1 0 1 0 0 0 0 0 1 1 0 1 1 0 1 1 1 0 0 1 0 1 1 0 1 1 0 0 1 1 1 1 1 0 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 0 0|1 0 1 0 0 0 0 1 0 1 1 0 1 1 1 0 1 1 1 0 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 0 1 1 1 0 1 1 0 0 0 1 1 1 0 0 0 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 1 0 1 0 1 0 0 0 1 1 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 1 1 1 0 0 1 0 1 1 1 1 0 0 1 1 0 1 0 1 1 0 0 0 1 1 1 0 1 1 1 1 1 0 0 0 1 1 0 1 0 1 0 1 1 0 0 1 1 0 1 0 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0|0 0 1 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 0 1 1 0 1 1 0 1 0 0 0 0 1 1 0 0 1 0 1 1 1 1 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 1 1 1 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 1 0 1 1 1 0 1 1 1 0 1 1 1 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 0 0 1 1 0 1 0 1 1 0 0|1 1 1 0 1 0 0 0 0 1 1 1 0 1 1 0 0 0 0 1 1 0 1 1 0 0 1 1 0 1 0 1 0 1 1 0 0 0 1 0 0 1 1 1 0 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 1 0 1 1 0 1 1 0 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 0 0 0 1 0 1 1 0 0 0 0 1 0 1 0 1 0 0 1 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 1 1 1 0 0 0 0 1 0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 1 0 1 0 0 0|1 1 1 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 1 0 0 0 1 0 1 0 1 0 1 1 0 1 0 0 1 1 1 1 1 1 1 0 0 1 1 0 0 1 1 0 1 0 0 1 0 0 1 1 1 0 1 0 1 1 1 0 1 0 0 0 1 0 0 1 1 1 1 0 1 1 0 1 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 1 1 1 1 1 1 0 1 0 1 1 1 1 1 0 1 0 1 0 0 0 0 1 0 1 1 0 1 1 1 0 0 0 0 1 1 1 1 1 1 0 1 1 1 0 0 1 0 1 1 0 0 0 0 1 0 1 0 0 0 0 1 1 1 0 0|0 1 0 1 0 0 0 0 1 0 0 1 1 1 1 1 0 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 1 0 0 0 0 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 0 1 1 1 1 1 1 0 0 1 1 1 1 1 0 1 1 0 1 1 1 1 0 1 0 0 0 1 0 0 1 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 0 1 1 0 1 1 0 1 0 1 1 0 1 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 1 1 0 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 1 1 1 0 1 0 1 0 1 1 0 0 1 1 0 0 0|1 0 0 1 0 1 0 0 0 0 0 0 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 0 0 1 1 0 1 1 0 0 1 1 1 1 0 0 0 1 0 1 1 0 0 1 1 1 0 0 0 1 1 0 1 1 0 0 1 1 0 1 1 1 0 0 1 0 1 1 1 0 1 0 1 1 0 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 1 1 0 0 1 1 1 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 1 1 1 0 1 0 1 0 0 1 1 0 1 1 1 0 0 0 0 1 0 1 1 1 0 0 1 0 1 1 1 0 0 1 0 0 1 0 0 0|0 1 0 1 1 0 1 1 1 0 1 0 1 0 1 1 0 1 0 1 0 0 0 0 1 1 1 1 0 1 0 1 1 0 1 1 1 1 1 0 1 1 0 1 0 0 0 1 1 0 0 1 1 1 1 1 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 1 0 1 1 1 1 0 0 1 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 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 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 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 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 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 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 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 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: 2006-04-03
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