Bounds on the minimum distance of additive quantum codes

Bounds on [[38,30]]₂

Construction

Construction of a [[38,30,3]] quantum code:
[1]:  [[85, 77, 3]] quantum code over GF(2^2)
     cyclic code of length 85 with generating polynomial w*x^84 + w^2*x^83 + w^2*x^81 + w^2*x^79 + w*x^78 + w^2*x^76 + w*x^74 + w^2*x^72 + w^2*x^71 + w^2*x^70 + w*x^69 + w^2*x^68 + x^67 + w^2*x^66 + w*x^64 + x^63 + x^62 + w^2*x^60 + x^59 + w^2*x^58 + w*x^57 + x^56 + x^54 + x^53 + w^2*x^52 + w*x^49 + w*x^46 + x^45 + w*x^44 + x^43 + x^42 + x^41 + x^40 + w*x^39 + w^2*x^38 + x^37 + x^36 + w^2*x^35 + w^2*x^34 + x^33 + w*x^30 + w*x^29 + w^2*x^28 + w^2*x^27 + x^26 + w^2*x^25 + x^24 + w*x^22 + x^21 + w^2*x^20 + w^2*x^19 + w*x^18 + x^17 + w*x^16 + w*x^15 + w^2*x^13 + x^12 + w^2*x^10 + w^2*x^9 + w^2*x^7 + x^6 + x^5 + x^4 + 1
[2]:  [[38, 30, 3]] quantum code over GF(2^2)
     Shortening of [1] at { 1, 2, 3, 5, 8, 9, 12, 14, 16, 19, 21, 22, 24, 27, 28, 30, 31, 32, 34, 35, 36, 37, 38, 39, 41, 42, 46, 47, 48, 51, 52, 56, 57, 60, 61, 62, 66, 67, 70, 71, 72, 73, 76, 77, 78, 81, 84 }

    stabilizer matrix:

      [1 0 0 0 1 1 1 1 1 0 0 1 1 1 0 1 0 1 0 1 1 1 1 0 0 0 1 1 1 1 1 0 0 1 1 1 0 1|0 0 0 0 1 0 0 1 0 0 1 1 0 1 1 1 1 0 0 0 0 1 1 0 0 1 1 1 0 0 1 1 0 1 0 1 1 0]
      [0 0 0 0 1 0 0 1 0 0 1 1 0 1 1 1 1 0 0 0 0 1 1 0 0 1 1 1 0 0 1 1 0 1 0 1 1 0|1 0 0 0 0 1 1 0 1 0 1 0 1 0 1 0 1 1 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 1 0 1 1]
      [0 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 0 0 1 0 0 1 1 0 0 0 0 1 1 1 0 1 1 1 1 0 0 0|0 0 0 0 1 1 0 1 1 1 0 1 1 1 1 1 0 0 1 1 1 1 0 1 1 1 1 0 1 0 0 1 0 0 1 0 0 1]
      [0 0 0 0 1 1 0 1 1 1 0 1 1 1 1 1 0 0 1 1 1 1 0 1 1 1 1 0 1 0 0 1 0 0 1 0 0 1|0 1 0 0 0 1 1 1 0 1 1 0 0 1 1 0 0 0 0 1 1 0 1 1 1 1 1 1 0 1 0 0 1 1 0 0 0 1]
      [0 0 1 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 0 1 1 0 0 0 0 0 1 0 1 1 1 0 1 1|0 0 0 0 0 1 1 0 0 0 0 1 1 0 1 0 0 0 1 1 1 1 1 1 1 0 1 0 1 1 0 0 0 0 0 0 1 0]
      [0 0 0 0 0 1 1 0 0 0 0 1 1 0 1 0 0 0 1 1 1 1 1 1 1 0 1 0 1 1 0 0 0 0 0 0 1 0|0 0 1 0 0 1 1 0 1 0 0 0 1 1 1 0 1 1 1 1 1 0 1 0 0 0 1 0 1 1 1 0 1 1 1 0 0 1]
      [0 0 0 1 0 0 1 0 1 0 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 0 0 0 1 1 1 1 1 1 0 1 1 0|0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 0 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 1 1 0 0 1]
      [0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 0 0 0 1 0 1 0 0 1 1 0 0 1 1 0 1 1 1 0 0 1|0 0 0 1 0 1 1 0 1 1 1 1 0 0 0 1 0 0 1 0 0 1 0 0 1 1 1 0 1 0 0 1 0 0 1 1 1 1]

last modified: 2006-04-07

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.