Bounds on the minimum distance of additive quantum codes

Bounds on [[25,7]]₂

Construction

Construction of a [[25,7,5]] quantum code:
[1]:  [[21, 7, 5]] quantum code over GF(2^2)
     cyclic code of length 21 with generating polynomials [
x^20 + w^2*x^19 + w^2*x^18 + w*x^15 + w^2*x^14 + w^2*x^12 + x^10 + x^9 + w*x^7 + w*x^6 + w*x^5 + 1,
w*x^20 + w^2*x^19 + w*x^18 + x^16 + x^15 + w*x^14 + x^13 + w*x^12 + x^11 + w*x^10 + w^2*x^9 + w
]
[2]:  [[25, 7, 5]] quantum code over GF(2^2)
     ExtendCode [1] by 4

    stabilizer matrix:

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

last modified: 2005-06-24

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 even 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.