Bounds on the minimum distance of additive quantum codes

Bounds on [[212,141]]₂

Construction

Construction of a [[212,141,10]] quantum code:
[1]:  [[205, 141, 10]] quantum code over GF(2^2)
     QuasiTwistedCyclicCode of length 205 and constant w^2 with generators: x^173 + w^2*x^172 + w*x^168 + x^166 + w^2*x^164 + w*x^159 + x^157 + w*x^156 + w^2*x^153 + w*x^152 + w^2*x^151 + x^150 + x^149 + x^148 + x^147 + x^146 + x^145 + w^2*x^143 + x^142 + w^2*x^140 + w*x^139 + w*x^138 + x^137 + w^2*x^136 + w^2*x^135 + x^134 + x^133 + w^2*x^131 + w*x^130 + w*x^129 + w*x^126 + w*x^125 + x^123 + w^2*x^122 + x^120 + x^119 + x^118 + w*x^117 + w*x^116 + w^2*x^114 + w^2*x^113 + x^112 + w^2*x^111 + w^2*x^110 + x^107 + w*x^106 + w^2*x^105 + x^104 + x^103 + x^102 + x^101 + x^100 + x^99 + w*x^98 + x^97 + w*x^96 + w*x^94 + w*x^93 + w^2*x^92 + w^2*x^91 + x^90 + w*x^89 + w*x^88 + w*x^87 + x^86 + w*x^85 + w^2*x^84 + w^2*x^83 + w^2*x^82 + x^81 + x^80 + w*x^79 + x^77 + x^76 + w^2*x^75 + w^2*x^74 + x^73 + w*x^72 + w^2*x^71 + x^70 + w*x^69 + w*x^68 + w*x^67 + w*x^66 + x^63 + w*x^62 + x^61 + x^60 + w*x^59 + w^2*x^57 + x^56 + x^55 + w*x^54 + w^2*x^53 + x^51 + w^2*x^50 + w^2*x^48 + x^47 + x^44 + w*x^43 + x^42 + x^40 + w*x^39 + w*x^38 + w^2*x^37 + w*x^36 + x^35 + w*x^34 + x^33 + x^31 + x^30 + x^28 + w*x^27 + w^2*x^26 + x^25 + w*x^24 + w^2*x^23 + w^2*x^22 + w^2*x^21 + w*x^20 + x^17 + x^16 + x^14 + x^9 + x^7 + x^5 + w^2*x + w
[2]:  [[212, 141, 10]] quantum code over GF(2^2)
     ExtendCode [1] by 7

    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 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 1 1 0 0 0 0 0 0 1 0 1 1 1 1 0 0 0 1 0 0 0 0 1 0 1 0 0 1 1 0 0 1 0 1 0 0 1 0 0 0 1 1 1 1 0 1 1 0 1 1 0 0 0 1 0 0 1 1 1 1 0 1 1 1 0 1 1 1 1 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 1 1 0 1 1 0 1 1 0 0 0 0 1 0 0 1 1 0 0 1 1 1 0 0 0 1 1 0 1 1 1 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 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 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 1 1 0 0 1 0 0 0 1 1 0 1 1 0 1 1 0 1 1 1 1 0 1 1 1 0 1 1 1 0 0 1 0 0 0 1 0 0 1 1 1 0 0 1 1 1 1 1 0 0 1 1 1 1 1 0 1 1 0 0 1 1 0 1 1 1 0 0 0 1 1 1 1 1 1 0 0 1 1 0 1 1 1 1 1 1 1 1 1 0 1 1 0 0 0 0 1 0 0 0 1 1 1 1 1 0 0 1 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 0 0 1 0 0 1 1 1 1 1 1 0 1 0 0 0 1 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0]
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last modified: 2024-10-16

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.