Bounds on the minimum distance of additive quantum codes

Bounds on [[249,195]]₂

Construction

Construction of a [[249,195,8]] quantum code:
[1]:  [[247, 199, 8]] quantum code over GF(2^2)
     constacyclic code generated by x^223 + w^2*x^222 + w*x^221 + x^220 + w^2*x^219 + w^2*x^216 + x^215 + w^2*x^213 + w*x^212 + w*x^211 + w^2*x^210 + w*x^208 + x^207 + w*x^205 + w*x^204 + x^203 + w^2*x^202 + w*x^200 + w^2*x^199 + x^196 + w*x^195 + w*x^194 + w*x^192 + w^2*x^189 + x^187 + w*x^186 + w^2*x^185 + w*x^184 + x^182 + w^2*x^180 + w*x^179 + w^2*x^177 + x^176 + w*x^175 + w*x^174 + w^2*x^171 + w*x^170 + w^2*x^168 + w*x^167 + x^166 + x^165 + w^2*x^164 + x^163 + x^161 + w^2*x^160 + w*x^158 + w^2*x^157 + x^156 + w*x^155 + w^2*x^154 + x^153 + x^150 + w^2*x^149 + w*x^148 + w*x^147 + w*x^145 + w^2*x^143 + x^141 + w^2*x^140 + w^2*x^139 + w*x^138 + w*x^136 + w*x^134 + x^132 + w^2*x^131 + w*x^130 + w^2*x^129 + w^2*x^128 + x^127 + w*x^124 + w*x^121 + w^2*x^120 + w^2*x^119 + w*x^118 + w^2*x^117 + w^2*x^116 + w^2*x^114 + x^113 + w*x^111 + w*x^110 + x^109 + w^2*x^108 + w*x^104 + x^102 + x^101 + w*x^99 + w^2*x^98 + x^97 + x^96 + w^2*x^94 + x^93 + w*x^92 + w^2*x^91 + x^88 + w*x^87 + w*x^86 + w*x^85 + w^2*x^84 + w^2*x^83 + x^79 + w^2*x^77 + x^76 + x^75 + x^73 + x^72 + w^2*x^71 + w^2*x^70 + x^69 + w*x^68 + w*x^67 + w^2*x^66 + x^65 + x^64 + w*x^62 + w^2*x^59 + w*x^58 + w^2*x^57 + x^54 + w*x^53 + x^52 + w*x^51 + w^2*x^50 + w^2*x^49 + x^48 + w*x^46 + x^45 + w^2*x^43 + w*x^42 + w^2*x^41 + w*x^40 + w^2*x^38 + x^37 + w*x^35 + x^34 + w^2*x^33 + w^2*x^32 + w*x^30 + w^2*x^28 + w*x^27 + w*x^26 + w*x^25 + x^23 + w*x^22 + w^2*x^21 + w*x^20 + w^2*x^19 + x^18 + x^17 + w^2*x^16 + w*x^15 + w*x^14 + x^13 + x^11 + x^9 + w*x^7 + x^6 + w^2*x^3 + w^2*x^2 + x + w with shift constant w
[2]:  [[247, 195, 8]] quantum code over GF(2^2)
     Subcode of [1]
[3]:  [[249, 195, 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 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 1 1 0 1 1 0 1 0 1 1 0 1 1 1 0 0 1 0 0 0 1 1 0 1 0 0 1 0 1 0 0 1 1 0 1 0 0 1 1 1 1 0 0 0 1 0 0 0 1 0 1 1 0 1 1 1 0 0 1 1 0 1 1 0 0 1 0 0 0 0 0 1 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0 1 1 0 1 0 0 0 0 1 1 1 0 1 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 1 0 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 0 1 1 1 0 0 1 0 0 1 0 1 0 1 1 0 0 0 0 1 0 1 1 1 0 0 0 1 0 1 0 1 0 1 1 0 1 1 1 0 1 1 0 1 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 1 1 0 0 1 0 0 1 0 1 0 1 0 0 1 1 1 1 0 1 0 1 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 0 0 1 0 1 0 1 0 0 1 1 1 0 1 0 1 0 0 1 0 1 0 0 0 0 1 1 1 0 0 1 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 0 1 1 0 1 1 1 0 0 1 1 0 0 0 0 0 1 1 0 0 0 1 1 0 1 1 0 1 1 0 0 0 0 0 0 1 1 1 0 1 1 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 1 1 0 0 1 1 0 1 1 0 0 1 1 0 1 1 1 1 0 1 0 0 1 0 0 0 0 1 1 0 0 1 0 1 0 0 1 0 1 0 1 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 1 0 0 1 0 0 1 0 1 1 1 1 0 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 1 1 0 0 1 0 0 1 0 1 0 1 0 0 1 1 1 1 0 1 0 1 0 0 0 0 1 0 0 0 1 1 1 0 0 1 1 0 0 1 0 1 0 1 0 0 1 1 1 0 1 0 1 0 0 1 0 1 0 0 0 0 1 1 1 0 0 1 1 1 1 1 0 1 1 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 0 1 1 0 1 1 1 0 0 1 1 0 0 0 0 0 1 1 0 0 0 1 1 0 1 1 0 1 1 0 0 0 0 0 0 1 1 1 0 1 1 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 1 1 0 0 1 1 0 1 1 0 0 1 1 0 1 1 1 1 0 1 0 0 1 0 0 0 0 1 1 0 0 1 0 1 0 0 1 0 1 0 1 0 0 0 0 0 0 1 0 0 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 0 1 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 1 1 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 1 1 1 1 0 0 1 1 1 1 0 1 0 1 1 0 1 0 0 1 0 1 1 1 1 0 0 1 0 0 1 1 1 0 1 0 0 0 1 1 1 0 0 1 0 0 0 1 1 1 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 1 1 1 1 0 0 1 0 0 0 1 1 1 1 0 0 1 0 1 0 1 1 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 0 1 1 0 1 0 0 0 1 0 0 1 1 0 1 1 0 0 1 1 0 1 0 1 1 0 0 0 1 1 1 0 0 1 0 0 1 0 1 1 0 0 1 1 0 1 1 1 1 1 1 0 0 0 1 0 1 1 0 0 0 1 0 0 0 0 1 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 1 1 1 0 1 0 0 1 1 1 1 1 0 1 0 0 0 1 0 0 0 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 1 0 1 0 0 1 0 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 0 0 1 0 0 1 1 1 0 0 0 1 1 0 1 0 1 0 1 0 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 0 1 0 1 1 1 1 1 1 0 1 1 1 1 0 0 1 1 0 1 0 1 0 1 1 1 1 0 1 0 0 0 1 1 0 0 0 1 1 1 1 0 0 0 0 1 1 0 0 1 0 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 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 1 1 0 0 1 0 1 1 0 1 1 1 0 1 0 0 0 0 0 1 1 0 1 0 0 1 1 1 1 1 1 0 1 0 0 1 0 1 1 1 1 1 0 0 0 1 0 1 1 1 0 0 0 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1 1 0 1 0 1 0 0 0 0 1 1 1 0 1 1 1 1 0 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 1 1 0 0 1 1 0 0 0 1 0 0 1 1 0 0 1 0 1 0 0 0 0 1 1 0 1 0 1 1 0 1 0 0 0 1 1 1 0 0 1 0 1 0 0 0 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 1 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 1 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 1 0 0 0 1 0 1 0 0 0 0 1 0 0 1 1 1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 0 0 1 1 1 0 0 1 1 0 1 1 1 0 0 0 1 0 1 1 0 1 1 1 1 0 0 1 0 0 0 1 0 0 0 1 1 1 0 1 1 1 0 1 1 0 0 0 1 1 0 0 1 1 1 1 1 1 0 1 0 1 1 0 1 1 1 0 0 0 1 0 0 0 1 1 0 1 1 1 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 1 0 1 1 1 1 0 0 1 1 1 0 1 1 0 0 0 1 0 1 0 1 1 1 0 0 1 1 1 1 0 0 0 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 1 1 1 1 0 1 0 0 1 1 1 1 1 0 1 0 0 0 1 0 0 0 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 1 0 1 0 0 1 0 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 0 0 1 0 0 1 1 1 0 0 0 1 1 0 1 0 1 0 1 0 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 0 1 0 1 1 1 1 1 1 0 1 1 1 1 0 0 1 1 0 1 0 1 0 1 1 0 0 0 1 1 1 1 1 0 0 1 1 1 0 0 1 0 1 0 1 1 0 1 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 1 1 1 1 0 1 0 0 1 1 1 1 1 0 1 0 0 0 1 0 0 0 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 1 1 0 0 0 0 1 0 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 1 0 1 0 0 1 0 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 0 1 0 0 0 1 0 0 1 1 1 0 0 0 1 1 0 1 0 1 0 1 0 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 0 1 0 1 1 1 1 1 1 0 1 1 1 1 0 0 1 1 0 1 0 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 1 1 0 1 1 1 0 1 0 0 1 0 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 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 0 1 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 1 1 0 0 1 0 1 1 0 1 1 1 0 1 0 0 0 0 0 1 1 0 1 0 0 1 1 1 1 1 1 0 1 0 0 1 0 1 1 1 1 1 0 0 0 1 0 1 1 1 0 0 0 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1 1 0 1 0 1 0 0 0 0 1 1 1 0 1 1 1 1 0 0 1 1 0 1 0 0 1 0 0 1 0 0 1 0 1 1 0 0 1 1 0 0 0 1 0 0 1 1 0 0 1 0 1 0 0 1 1 1 1 0 1 0 1 0 0 1 1 0 1 0 0 1 0 0 0 1 0 1 1 0 0]
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last modified: 2024-06-09

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.