Bounds on the minimum distance of additive quantum codes

Bounds on [[181,0]]₂

Construction

Construction of a [[181,0,22]] quantum code:
[1]:  [191, 96, 27] Cyclic Linear Code over GF(2)
     cyclic code of length 191 with generating polynomial x^95 + x^94 + x^93 + x^92 + x^91 + x^90 + x^89 + x^84 + x^83 + x^82 + x^79 + x^78 + x^76 + x^75 + x^73 + x^70 + x^68 + x^64 + x^63 + x^62 + x^58 + x^55 + x^53 + x^51 + x^50 + x^48 + x^47 + x^46 + x^45 + x^41 + x^40 + x^38 + x^36 + x^35 + x^34 + x^31 + x^30 + x^29 + x^27 + x^26 + x^23 + x^19 + x^18 + x^16 + x^11 + x^7 + 1
[2]:  [192, 96, 28] Linear Code over GF(2)
     ExtendCode [1] by 1
[3]:  [[192, 0, 28]] CSS self-dual quantum code over GF(2^2)
     CSS code using  [2] and  [2]
[4]:  [[182, 0, 23]] CSS self-dual quantum code over GF(2^2)
     shortening & puncturing CSS code  [3]
[5]:  [[181, 0, 22]] self-dual quantum code over GF(2^2)
     Puncturing of [4] at { 182 }

    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 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 1 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 1 1 0 0 0 1 1 1 1 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 1 1 1 0 1 0 1 1 1 1 0 1 0 1 0 0 0 1 1 1 1 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
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last modified: 2022-08-08

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 necessarily 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.