Bounds on the minimum distance of additive quantum codes
Bounds on [[78,1]]2
lower bound: | 18 |
upper bound: | 27 |
Construction
Construction of a [[78,1,18]] quantum code:
[1]: [[79, 0, 19]] self-dual quantum code over GF(2^2)
cyclic code of length 79 with generating polynomial x^78 + x^76 + x^73 + x^70 + x^67 + x^65 + x^62 + x^61 + x^60 + x^59 + x^57 + x^56 + x^55 + x^51 + x^50 + x^49 + x^48 + x^45 + x^44 + x^43 + x^41 + x^38 + x^36 + x^35 + x^34 + x^31 + x^30 + x^29 + x^28 + x^24 + x^23 + x^22 + x^20 + x^19 + x^18 + x^17 + x^14 + x^12 + x^9 + x^6 + x^3 + x + w
[2]: [[78, 1, 18]] quantum code over GF(2^2)
Shortening of the stabilizer code of [1] at 79
stabilizer matrix:
[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 0 0 0 0 0 1 1 1 0 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 1 0 0 0 0 1 0 1 0 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 1 1 0 0 0 1 1 0 0 1 0 0 0 1 1 1 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 1 1 1 1 0 0 1 1 0 0 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 0 1 1 1 0 1 1 0 1 0 1 0 1 0 1 0 1 1 0 1 1 1 0 1 0 1 0 0 0 0 1 1 0 0 0 1 0 1 0 0 1 1 0 0 1 1 1 1 0 1]
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[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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|1 1 1 1 1 1 0 1 0 0 0 0 1 1 1 0 1 1 1 1 0 1 1 0 0 0 1 1 1 1 1 0 0 1 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0 0 0 1 1 0 1 1 1 1 0 1 1 1 0 0 0 0 1 0 1 1 1 1 1 1 0 0 0]
last modified: 2011-06-21
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.
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Markus Grassl
(codes@codetables.de).
Last change: 23.10.2014