Bounds on the minimum distance of additive quantum codes

Bounds on [[211,0]]₂

Construction

Construction of a [[211,0,25]] quantum code:
[1]:  [223, 112, 31] Cyclic Linear Code over GF(2)
     cyclic code of length 223 with generating polynomial x^111 + x^110 + x^109 + x^106 + x^100 + x^96 + x^95 + x^94 + x^92 + x^85 + x^83 + x^82 + x^81 + x^79 + x^78 + x^77 + x^72 + x^66 + x^64 + x^62 + x^60 + x^58 + x^57 + x^56 + x^51 + x^50 + x^45 + x^44 + x^42 + x^41 + x^40 + x^38 + x^36 + x^35 + x^32 + x^31 + x^29 + x^27 + x^25 + x^22 + x^21 + x^19 + x^18 + x^17 + x^16 + x^13 + x^10 + x^9 + x^5 + x^3 + 1
[2]:  [224, 112, 32] Linear Code over GF(2)
     ExtendCode [1] by 1
[3]:  [[224, 0, 32]] CSS self-dual quantum code over GF(2^2)
     CSS code using  [2] and  [2]
[4]:  [[212, 0, 26]] CSS self-dual quantum code over GF(2^2)
     shortening & puncturing CSS code  [3]
[5]:  [[211, 0, 25]] self-dual quantum code over GF(2^2)
     Puncturing of [4] at { 212 }

    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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 0 0 1 1 1 0 0 1 0 0 0 1 1 0 0 0 1 0 1 0 1 0 1 0 0 0 1 1 1 0 1 0 0 1 0 0 1 0 1 1 1 1 0 1 1 0 1 0 0 0 0 1 0 0 0 1 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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.