Bounds on the minimum distance of additive quantum codes
Bounds on [[79,39]]2
lower bound: | 9 |
upper bound: | 14 |
Construction
Construction of a [[79,39,9]] quantum code:
[1]: [[79, 40, 9]] quantum code over GF(2^2)
cyclic code of length 79 with generating polynomial x^78 + x^77 + x^75 + w*x^74 + w^2*x^72 + w*x^71 + w^2*x^70 + x^69 + w*x^68 + x^65 + w^2*x^63 + w^2*x^62 + w*x^59 + x^58 + w*x^57 + x^56 + x^55 + x^54 + x^53 + w^2*x^52 + w*x^51 + w*x^50 + x^48 + w*x^47 + w^2*x^45 + w*x^43 + w^2*x^42 + w^2*x^41 + x^40 + w*x^39 + x^38 + x^36 + w*x^35 + w^2*x^32 + x^31 + w^2*x^30 + w^2*x^29 + x^27 + x^26 + w*x^24 + x^23 + w^2*x^22 + w*x^21 + w*x^20 + w^2*x^18 + w^2*x^17 + w*x^15 + w^2*x^14 + w*x^12 + w*x^11 + x^10 + x^9 + x^8 + w*x^6 + x^5 + w^2*x^4 + x^3 + x^2 + w*x + w
[2]: [[79, 39, 9]] quantum code over GF(2^2)
Subcode of [1]
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 1 1 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0 1 1 0 0 1 0 0 1 1 1 0 1 1 0 1 1 0 1 0 1 0|0 1 0 1 0 0 0 1 1 0 1 0 0 0 1 0 0 1 1 1 1 0 1 1 0 0 0 1 0 0 1 1 0 0 1 0 0 1 0 0 1 0 1 1 0 0 1 0 1 0 0 0 1 0 1 0 1 0 1 0 0 0 0 1 0 1 0 1 1 1 1 0 1 1 0 1 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 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 1 0 0 1 1 0 0 0 1 0 1 0 0 0 1 1 1 1 0 1 1 1 0 0 0 0 1 0 1 1 0|1 0 0 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 1 1 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 1 0 0 1 0 1 1 0 1 1 0 1 0 0 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 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 1 0 1 0 0 1 0 0 1 1 0 0 0 1 0 1 0 0 0 1 1 1 1 0 1 1 1 0 0 0 0 1 0 1 1|0 1 0 0 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 1 1 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 1 1 1 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 1 0 0 1 0 1 1 0 1 1 0 1 0 0 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 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 0 0 1 1 0 0 0 1 1 0 1 1 1 0 0 1 1 0 1 0 1 0 0 0 0 0 0 1 1 1 0 1 1 1 1|0 1 1 1 0 0 0 1 0 1 1 0 1 0 0 0 1 1 1 1 0 1 0 1 0 1 0 1 0 1 1 1 0 1 1 0 0 0 1 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 1 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 0]
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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 1 0 1 1 0 1 1 1 1 0 0 0 0 1 1 0 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 1 0 1 0 0 0 1 1 0|0 1 0 1 0 1 1 1 1 1 1 1 0 1 1 0 0 0 0 0 1 0 0 1 0 0 1 1 0 0 0 1 0 0 0 1 1 0 1 1 0 0 1 1 0 1 1 0 1 1 0 0 1 0 0 1 0 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 0 1 0]
last modified: 2020-08-24
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