Bounds on the minimum distance of additive quantum codes
Bounds on [[51,30]]2
lower bound: | 5 |
upper bound: | 7 |
Construction
Construction of a [[51,30,5]] quantum code:
[1]: [[93, 73, 5]] quantum code over GF(2^2)
quasicyclic code of length 93 stacked to height 2 with 6 generating polynomials
[2]: [[50, 30, 5]] quantum code over GF(2^2)
Shortening of [1] at { 1, 4, 5, 6, 7, 11, 12, 14, 15, 16, 17, 20, 22, 27, 29, 33, 40, 41, 42, 44, 45, 48, 52, 53, 54, 61, 62, 64, 66, 67, 68, 69, 71, 73, 74, 76, 77, 82, 83, 84, 87, 92, 93 }
[3]: [[51, 30, 5]] quantum code over GF(2^2)
ExtendCode [2] by 1
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 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 1 1 0 1 0 1 0 0 0 0 1 1 0 0|1 1 1 1 0 1 1 1 0 0 0 1 1 1 1 1 0 1 1 0 0 0 1 1 1 1 1 0 1 1 0 1 0 0 0 0 1 1 0 0 1 1 0 1 1 1 1 1 0 1 0]
[0 1 0 0 0 0 0 0 0 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 0 1 1 1 1 1 1 1 1 1 0 1 1 1 0 0 0|1 0 1 0 0 0 1 1 0 1 1 1 1 0 1 1 0 1 0 1 1 1 1 0 0 0 1 1 1 0 1 0 1 1 1 1 0 0 0 1 0 0 1 1 1 1 0 1 0 1 0]
[0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0|0 0 1 0 0 1 0 0 1 0 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 1 0 1 0 0 0 1 1 1 0 1 0 1 0 1 1 0 0 0 1 0 0 1 1 1 0]
[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 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 0 1 0 0 0|0 0 1 1 1 1 0 1 1 1 0 0 0 1 1 1 0 1 0 0 1 1 0 1 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 1 1 1 0 0]
[0 0 0 0 1 0 0 0 0 0 0 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 1 0 0 1 1 1 0 1 0 1 1 1 1 0|1 0 1 1 0 1 0 0 1 0 1 0 0 1 0 1 1 1 0 0 0 1 0 1 0 1 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0]
[0 0 0 0 0 1 0 0 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 0 1 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0|0 0 0 0 0 1 1 1 0 1 0 0 0 0 1 1 1 1 1 1 0 1 1 0 0 1 0 1 1 0 0 0 0 1 0 1 1 0 1 0 1 0 0 0 0 0 1 0 1 0 0]
[0 0 0 0 0 0 1 0 0 0 0 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 1 1 0 1 0 1 1 0 0 1 0 1 1 0 0|1 1 0 1 1 0 0 0 1 0 1 0 1 1 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 1 0 1 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 0 1 1 0]
[0 0 0 0 0 0 0 1 0 0 0 0 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 1 0 0 1 1 1 0 0 0 1 0|0 0 0 1 1 0 1 0 0 1 1 0 0 0 1 0 1 1 0 0 1 1 1 1 0 1 0 0 1 0 0 1 1 0 0 1 1 1 0 1 1 0 0 1 1 0 1 1 0 1 0]
[0 0 0 0 0 0 0 0 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 1 1 1 0 1 0 0 1 1 0 1 1 0 1 1 0|0 1 0 1 0 0 1 0 0 0 0 0 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 1 1 0 1 0 1 0 1 1 1 0 1 0 0 1 0 0 1 0 1 1 0]
[0 0 0 0 0 0 0 0 0 0 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 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0|0 0 0 0 1 1 1 0 0 1 0 1 0 0 1 0 0 1 0 1 1 1 0 1 0 0 0 0 1 1 0 0 1 1 1 1 1 1 1 1 1 0 0 0 1 1 0 0 1 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 0 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 1 1 0 0 0 0 1 1 0 1 0|1 1 0 0 1 0 0 0 1 1 1 0 1 1 0 0 1 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 1 1 0 1 0 1 0 0 0 1 1 0 1 0 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 0 0 0 1 1 0 1 1 1 1 0 1 1 0 1 0 0 0 1 1 0 1 0 0 1 0 0 0|1 0 0 1 1 0 1 0 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 0 1 1 1 1 1 0 1 0 0 1 1 1 1 0 0 0 1 1 0 0 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 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 1 1 0 1 1 0 1 0 0|1 1 1 0 1 0 1 1 1 1 1 0 0 1 0 0 0 0 1 0 1 0 1 0 1 1 0 1 1 1 0 1 0 0 1 1 0 0 0 1 1 1 1 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 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 1 0 1 0 1 1 0 1 1 0 1 0 0 0 0|1 0 1 1 1 1 0 1 1 0 0 1 1 0 1 0 1 1 0 0 1 0 1 1 1 1 1 0 1 1 1 0 0 1 0 1 1 1 0 1 1 0 0 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 1 0 0 0 0 0 0 1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0|0 1 1 1 0 1 0 0 1 0 1 0 1 1 0 0 0 0 1 1 1 0 0 1 1 1 1 0 0 0 1 0 1 0 1 0 1 1 0 0 0 0 1 0 1 0 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 1 0 0 0 0 1 1 0 1 0 1 0 1 0 1 1 1 1 0 1 1 1 0 1 1 0 0 0 1 0|1 0 0 1 0 1 1 0 0 1 1 1 0 1 1 1 0 1 0 0 0 0 0 0 1 1 0 0 0 1 0 0 1 1 0 1 1 1 0 0 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 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 0 1 1 0 0 0 0 0 1 1 0 1 0 1 1 0 0 1 0|0 0 1 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 0 1 0 1 1 1 0 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 0 1 0 0 0 1 1 0 0 1 0 1 1 0 1 0 1 1 0 0 1 1 1 1 1 1 1 0 0|1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 0 1 1 0 1 0 1 0 1 0 0 1 1 0 0 1 1 0 1 0 0 0 0 1 1 1 1 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 1 0 0 1 0 0 1 0 1 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0|0 1 1 1 1 0 1 0 1 1 1 1 1 1 1 0 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 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 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 1 0 0 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 0 0 0 1 0 1 0 0 0 1 1 0 0 0 1 1 0 0 1 0 0 1 1 0 0 1 0 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 0 0 0 0 0 0 0 0 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]
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