Bounds on the minimum distance of additive quantum codes
Bounds on [[85,72]]2
lower bound: | 4 |
upper bound: | 4 |
Construction
Construction of a [[85,72,4]] quantum code:
[1]: [[85, 73, 4]] quantum code over GF(2^2)
cyclic code of length 85 with generating polynomial w*x^84 + w^2*x^83 + w*x^81 + w*x^80 + x^79 + w^2*x^78 + w*x^76 + x^75 + w*x^74 + x^73 + x^71 + w^2*x^70 + w^2*x^69 + w*x^67 + w*x^66 + w^2*x^65 + x^64 + w^2*x^63 + x^62 + w*x^61 + w^2*x^60 + x^59 + w*x^57 + w^2*x^55 + w*x^54 + w*x^53 + w^2*x^52 + x^51 + x^50 + x^49 + w*x^48 + x^47 + w*x^46 + w*x^45 + x^44 + w*x^43 + x^42 + x^41 + x^40 + w^2*x^39 + w*x^38 + w*x^37 + w^2*x^36 + w*x^34 + x^32 + w^2*x^31 + w*x^30 + x^29 + w^2*x^28 + x^27 + w^2*x^26 + w*x^25 + w*x^24 + w^2*x^22 + w^2*x^21 + x^20 + x^18 + w*x^17 + x^16 + w*x^15 + w^2*x^13 + x^12 + w*x^11 + w*x^10 + w^2*x^8 + w*x^7 + x^6 + 1
[2]: [[85, 72, 4]] quantum code over GF(2^2)
Subcode of [1]
stabilizer matrix:
[1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 0 1 0 0 0 0 1 0 0 1 0 0 0 1 1 1 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 1 0 0 1 0 0 0 0 1 0 1 1 0 1 0 1 1 0 1 1 1 0|0 1 0 0 0 1 0 0 1 1 0 1 0 1 1 0 1 1 0 0 1 1 0 1 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 1 0 0 0 0 1 0 0 0 1 0 1 1 0 0 1 1 0 1 1 0 1 0 0 1 1]
[0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 0 1 1 0 1 0 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 1 1 1 0 1 1 1 0 0 1 1 1 0 0 1 0 1 0 1 0 1 0 0 1 1 1 0 0 1 1 1 0 0 1 0 1 0 0|1 1 0 0 1 0 1 0 0 0 1 1 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 1 1 0 0 0 1 0 1 0 0 1 1 0 0 0 0 1 1 1 0 1 1 1 0 1 1 1 1 1 0 1 1 0 1 1 0 1 1 1 1 1 0 1 1 1 0 1 1 1 0 1 0 1]
[0 0 1 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 1 0 0 1 1 1 0 1 0 1 0 1 1 0 1 1 1 1 0 1 0 1 1 0 0 1 1 0 1 0 1 1 1 1 0 1 1 0 1 0 1 0 1 1 1 0 0 1 1 1 0 1 0 0 1|1 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 0 1 1 0 0 0 1 1 1 0 0 0 1 0 1 1 1 1 1 1 1 0 1 1 1 0 0 1 0 1 1 1 0 1 1 0 1 0 0 1 0 1 1 0 1 1 1 0 1 0 0 1 1 1 0 1 1 1 1 1 1 1 0 1 0 0 1 1 0]
[0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 1 0 1 1 0 1 1 1 1 0 1 0 1 1 0 0 1 1 0 1 0 1 1 1 1 0 1 1 0 1 0 1 0 1 1 1 0 0 1 1 0 0 0 1 1 1 0 0 1|0 1 1 0 1 0 1 0 0 0 1 0 1 0 1 1 0 0 0 1 1 1 0 0 0 1 0 1 1 1 1 1 1 1 0 1 1 1 0 0 1 0 1 1 1 0 1 1 0 1 0 0 1 0 1 1 0 1 1 1 0 1 0 0 1 1 1 0 1 1 1 1 1 1 1 0 1 0 0 0 1 1 1 0 0]
[0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 1 0 1 0 1 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 0 1 1 0 1 1 0 1 1 0 0 0 0 1 1 0 0 1 1 0 0 1 0 0 0 1 0 1 0 1 1 1 1 0 0 1 1 1 0 0 1 0|0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 1 0 1 0 0 0 0 1 1 0 0 1 1 1 1 1 1 0 1 0 1 0 1 1 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1 1 0 1 0 1 0 1 1 1 1 1 1 0 0 1 1 0 0 0 0 1 0 1 1 1 0 1]
[0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 1 0 1 0 0 1 1 0 0 0 0 1 1 1 0 1 1 1 0 1 1 1 1 1 0 1 1 0 1 1 0 1 1 1 1 1 0 1 1 1 0 1 1 1 0 0 0 0 1 1 0 0 1 0 1 0 0 0 0 1 1 0 1 0|1 1 0 1 0 0 0 0 1 0 1 1 1 1 1 0 1 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 1 0 0 1 0 0 0 0 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 1 0 0 1 1 0 0 1 0 1 0 1 0 0 0 1 0 0 0 0 0 1 1 0 0 1 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 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 0 0 0 0 0 0 1 0 1]
[0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 0 0 1 0 1 0 0 1 1 0 0 0 0 1 1 1 0 1 1 1 0 1 1 1 1 1 0 1 1 0 1 1 0 1 1 1 1 1 0 1 1 1 0 1 1 1 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 1 0 0|1 1 1 0 1 0 0 0 0 1 0 1 1 1 1 1 0 1 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 1 0 0 1 0 0 0 0 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 1 0 0 1 1 0 0 1 0 1 0 1 0 0 0 1 0 0 0 0 0 1 1 1 1 0]
[0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 1 0 1 0 1 0 0 0 1 0 0 1 1 0 0 1 1 0 0 0 0 1 1 0 1 1 0 1 1 0 0 0 0 1 1 0 0 1 1 0 0 1 0 0 0 1 0 1 0 1 1 1 1 0 1 1 1 0 0 1|0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 1 1 0 1 0 0 0 0 1 1 0 0 1 1 1 1 1 1 0 1 0 1 0 1 1 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1 1 0 1 0 1 0 1 1 1 1 1 1 0 0 1 1 0 0 0 0 1 0 0 1 1]
[0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 1 0 1 1 0 1 1 1 1 0 1 0 1 1 0 0 1 1 0 1 0 1 1 1 1 0 1 1 0 1 0 1 0 1 1 1 0 0 1 1 0 0 0 1 1 1 0 0 1 0 0 0 1|1 0 1 0 0 0 1 0 1 0 1 1 0 0 0 1 1 1 0 0 0 1 0 1 1 1 1 1 1 1 0 1 1 1 0 0 1 0 1 1 1 0 1 1 0 1 0 0 1 0 1 1 0 1 1 1 0 1 0 0 1 1 1 0 1 1 1 1 1 1 1 0 1 0 0 0 1 1 1 0 0 0 1 1 0]
[0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 1 0 1 1 0 1 1 1 1 0 1 0 1 1 0 0 1 1 0 1 0 1 1 1 1 0 1 1 0 1 0 1 0 1 1 1 0 0 1 1 0 0 0 1 1 1 0 0 1 0 1 0 0 1 1 0|0 0 0 1 0 1 0 1 1 0 0 0 1 1 1 0 0 0 1 0 1 1 1 1 1 1 1 0 1 1 1 0 0 1 0 1 1 1 0 1 1 0 1 0 0 1 0 1 1 0 1 1 1 0 1 0 0 1 1 1 0 1 1 1 1 1 1 1 0 1 0 0 0 1 1 1 0 0 0 1 1 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 1 1 1 0 0 1 1 1 0 0 1 0 1 0 1 0 1 0 0 1 1 1 0 0 1 1 1 0 1 1 1 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 1 0 1 0 1 1 0 1 1 1 0 0 0 0|0 1 1 0 0 0 1 0 1 0 0 1 1 0 0 0 0 1 1 1 0 1 1 1 0 1 1 1 1 1 0 1 1 0 1 1 0 1 1 1 1 1 0 1 1 1 0 1 1 1 0 0 0 0 1 1 0 0 1 0 1 0 0 0 1 1 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 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 1 1 0 1 0 0 0 0 1 0 0 1 0 0 0 1 1 1 0 0 1 1 0 1 0 1 1 0 0 0 0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 1 0 1 1|0 1 0 1 1 0 0 1 0 0 0 1 0 0 1 1 0 1 0 1 1 0 1 1 0 0 1 1 0 1 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 1 0 0 0 0 1 0 0 0 1 0 1 1 0 0 1 1 1 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.
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Markus Grassl
(codes@codetables.de).
Last change: 23.10.2014