Bounds on the minimum distance of additive quantum codes
Bounds on [[111,72]]2
lower bound: | 8 |
upper bound: | 12 |
Construction
Construction of a [[111,72,8]] quantum code:
[1]: [[109, 73, 8]] quantum code over GF(2^2)
cyclic code of length 109 with generating polynomial x^108 + w*x^106 + x^105 + x^104 + x^103 + x^101 + w^2*x^100 + w^2*x^98 + w*x^97 + w*x^95 + w*x^94 + w*x^92 + x^89 + w^2*x^88 + w^2*x^86 + w^2*x^85 + x^84 + x^80 + x^79 + w^2*x^77 + w*x^76 + x^75 + w^2*x^74 + w*x^73 + w*x^72 + w*x^71 + x^70 + w^2*x^69 + w^2*x^68 + w*x^67 + x^66 + w*x^65 + x^64 + x^63 + w*x^62 + x^61 + w*x^60 + w^2*x^59 + w^2*x^58 + x^57 + w*x^56 + w*x^55 + w*x^54 + w^2*x^53 + x^52 + w*x^51 + w^2*x^50 + x^48 + x^47 + x^43 + w^2*x^42 + w^2*x^41 + w^2*x^39 + x^38 + w*x^35 + w*x^33 + w*x^32 + w*x^30 + w^2*x^29 + w^2*x^27 + x^26 + x^24 + x^23 + x^22 + w*x^21 + x^19 + x^18 + 1
[2]: [[109, 72, 8]] quantum code over GF(2^2)
Subcode of [1]
[3]: [[111, 72, 8]] quantum code over GF(2^2)
ExtendCode [2] by 2
stabilizer matrix:
[1 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 1 0 1 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 1 0 1 0 1 1 0 1 0 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 1 0 1 1 0 0 1 1 0 1 0 1 1 0 1 1 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 1 0 0 0 0 0 1 0 1 1 0 1 1 0 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 1 1 0 1 0 0 1 0 1 1 1 0 1 1 1 1 0 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0]
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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