Bounds on the minimum distance of additive quantum codes

Bounds on [[35,24]]₂

Construction

Construction of a [[35,24,4]] quantum code:
[1]:  [[35, 25, 4]] quantum code over GF(2^2)
     cyclic code of length 35 with generating polynomials [
w*x^34 + w^2*x^33 + w*x^32 + x^31 + x^29 + w^2*x^28 + w*x^26 + w*x^25 + w*x^23 + w^2*x^22 + w*x^21 + w^2*x^19 + w*x^17 + w^2*x^16 + w^2*x^15 + w^2*x^14 + w*x^13 + x^12 + x^11 + w*x^10 + w*x^9 + x^8 + x^7 + w^2*x^6 + x^5 + 1,
w^2*x^34 + x^33 + w^2*x^32 + w*x^31 + w*x^29 + x^28 + w^2*x^26 + w^2*x^25 + w^2*x^23 + x^22 + w^2*x^21 + x^19 + w^2*x^17 + x^16 + x^15 + x^14 + w^2*x^13 + w*x^12 + w*x^11 + w^2*x^10 + w^2*x^9 + w*x^8 + w*x^7 + x^6 + w*x^5 + w
]
[2]:  [[35, 24, 4]] quantum code over GF(2^2)
     Subcode of [1]

    stabilizer matrix:

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

last modified: 2005-06-29

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.