Bounds on the minimum distance of additive quantum codes
Bounds on [[72,51]]2
lower bound: | 5 |
upper bound: | 7 |
Construction
Construction of a [[72,51,5]] quantum code:
[1]: [[128, 105, 6]] quantum code over GF(2^2)
Construction from a stored generator matrix
[2]: [[127, 106, 5]] quantum code over GF(2^2)
Shortening of the stabilizer code of [1] at 128
[3]: [[72, 51, 5]] quantum code over GF(2^2)
Shortening of [2] at { 1, 4, 6, 10, 11, 12, 13, 16, 19, 23, 24, 25, 29, 30, 31, 33, 35, 36, 41, 42, 43, 46, 48, 49, 53, 56, 57, 61, 62, 65, 66, 67, 75, 77, 79, 80, 84, 85, 89, 92, 93, 94, 98, 102, 108, 109, 110, 113, 114, 116, 117, 121, 122, 124, 125 }
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 0 1 0 1 1 0 0 1 0 1 1 0 0 1 1 1 1 0 0 1 0 0 1 1 1 0 0 1 1 0 1 0 1 1 0 1 1 1 1 0 1 0 1 1 0 0 0 1 0 0|0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 1 0 1 1 1 0 0 1 0 0 0 1 0 0 1 0 1 1 0 0 0 0 1 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 0 0 0]
[0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 1 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 1 1 0 1 0 1 0 1 0 1 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0|0 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 1 0 1 1 1 1 0 0 0 1 0 1 1 0 0 0 0 1 1 1 1 1 0 1 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0]
[0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 0 0 1 0 0 1 1 1 1 0 0 0 1 0 0 0 0 1 1 1 0 1 0 1 1 1 1 0 1 1 0 1 1 1 1 1 0 0 0 0 0 0|0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 0 1 0 0 1 1 0 0 1 1 1 0 1 1 1 1 0 1 1 0 0 1 0 1 1 0 1 1 0 1 1 0 1 1 1 1 0 1 1 1 1 0 0 1 0 0 0 1 1 1 1 0 0 1 1 0 0]
[0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 1 1 1 1 0 1 1 1 1 1 1 0 1 0 0 0 1 0 0 1 0 0 1 1 0 1 1 1 1 0 0 0 1 0 1 1 0 1 0 1 1 1|0 0 0 0 0 1 0 1 0 0 0 1 0 1 1 0 1 1 0 1 0 0 1 0 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 0 1 1 1 0 1 0 1 1 0 1 1 1 1 0 0 0 1 1 0 0 1 1 0 1 1 0 1 0 0 1 1]
[0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 1 1 0 1 1 1 0 1 0 0 1 1 1 1 0 1 0 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 1|0 0 0 0 0 1 0 1 1 0 0 1 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 1 1 1 1 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 1 1 0 0 0 1 1 1 1 1 0 1 0 1]
[0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 1 1 1 0 0 0 1 1 1 1 1 0 0 0 1 0 1 1 0 1 1 1 1 0 0 0 0 1 1 1 0 1 0 0 1 1 0 1 0 0 1 1 0 0 0 1 0 0 1|0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 1 1 1 1 1 0 1 1 0 1 1 0 0 1 0 0 1 1 0 0 0 1 0 0 1 1 0 0 1 0 1 1 1 0 0 1 0 1 0 0 1 1 1 1 1 0]
[0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 1 1 1 0 1 0 1 0 0 0 0 1 0 0 1 1 1 1 1 1 0 1 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 0 1 1 1 1 1 0|0 0 0 0 0 1 0 1 1 0 0 0 1 1 0 1 1 0 0 0 0 1 1 1 1 0 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 1 0 1 0 1 0 0]
[0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 0 1 1 1 0 1 0 1 1 1 1 1 1 1 0 0 0 1 0 1 1 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 1 0 0|0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 1 1 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 1 1 0 0 1 1 0 1 0 0 0 0 1 1 0 1 1 1 1 0 1 0 1 0 0 1 0 0 0 0 1]
[0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 1 1 0 1 0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0|0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 0 1 0 0 1 1 0 0 1 1 1 0 1 1 1 1 0 1 1 0 0 1 0 1 1 0 1 1 0 1 1 0 1 1 1 1 0 1 1 1 1 0 0 1 0 0 0 1 1 1 1 0 0 1 1 0 0]
[0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 0 0 1 0 1 0 0 0 1 0 1 1 1 1 1 0 1 1 0 1 0 0 1 0 0 1 1 0 0 0 1 0 1 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 1|0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 1 1 1 1 0 0 0 1 1 0 1 1 1 1 1 0 0 0 1 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1 0]
[0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 1 0 0 1 1 1 0 1 1 0 0 1 1 0 1 0 0 1 1 1 0 1 0 0 0 1 0 1 0 0 0 1 0 0 1 0 1 0 1 1 0 0 1 0 1 0 0 0|0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 1 1 0 0 1 1 1 0 0 1 1 0 0 1 1 1 0 1 0 1 1 0 0 1 0 1 0 0 1 0 1]
[0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 0 0 1 1 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 1 1 0 0 1 1 0 1 0 0 0 0 1 1 0 1 1 1 1 0 1 0 1 0 0 1 0 0 0 0 1|0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 1 0 1 0 1 1 1 1 1 1 1 0 1 1 1 0 1 1 1 1 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 0 0 1 1 1 0 0 1 0 1 0 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 1 1 0 0 1 1 1 0 0 1 1 0 0 1 1 1 0 1 0 1 1 0 0 1 0 1 0 0 1 0 1|0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 1 1 1 1 0 0 0 1 1 0 1 1 1 1 1 0 0 0 1 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 1 1 1 0 0 0 1 1 0 1 0 1 0 0 1 1 1 0 1 0 1 0 1 1 1 1 0 1 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 1 1|0 0 0 0 0 1 0 0 1 0 0 0 1 0 1 0 1 1 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 1 1 1 1 0 1 0 1 1 1 0 1 0 0 0 1 0 1 0 1 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 0 1 0 0 0 1 0 0 1 1 0 0 1 0 1 1 1 1 1 0 0 1 1 1 1 0 0 1 1 0 1 0 0 1 0 0 1 1 1 0 1 1 1 1 0 1 0 1 1 1 0 0 0 1 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 1 1 0 1 0 0 1 1 1 1 0 0 0 1 1 0 1 0 0 1 1 0 1 0 0 1 1 0 1 1 0 0 1 1 1 1 1 0 0 1 1 0 1 0 1 1 0 1 1 0 0 1 1 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 0 1 1 1 1 0 0 1 1 0 0 1 0 0 1 1 0 1 0 0 0 1 1 0 1 1 0 0 1 0 0 0 1 1 0 1 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 1 0 1 1 1 0 0 1 1 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 0 1 0 0 1 1 1 0 1 1 0 1 1 0 1 0 1 0 1 0 1 0 1 1 0 1 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 0 0 0 1 0 0 1 1 0 0 1 0 1 0 1 0 1 1 1 0 1 1 0 1 0 0 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 1 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 1 1 1 1 0 0 1 0 0 1 1 1 1 1 1 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 1 0 1 1 1 0 0 0 1 0 0 1 0 1 1 1 0 0 1 0 1 1 1 1 0 1 0 0 0 1 0 0 1 1 0 1 0 1 1 0 1 0 0 1 0 1 1 1 1 0 0 1 0 0 1 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