Bounds on the minimum distance of additive quantum codes

Bounds on [[34,23]]₂

Construction

Construction of a [[34,23,4]] quantum code:
[1]:  [[34, 24, 4]] quantum code over GF(2^2)
     QuasiCyclicCode of length 34 stacked to height 2 with generating polynomials: w*x^16 + x^15 + w^2*x^14 + w*x^13 + w*x^12 + x^11 + w*x^10 + w*x^9 + w^2*x^8 + x^7 + w*x^6 + x^5 + 1,  x^16 + x^15 + w*x^13 + x^12 + w*x^11 + w^2*x^10 + w^2*x^9 + w^2*x^8 + w^2*x^7 + w*x^6 + x^5 + w*x^4 + x^2 + x + w,  w^2*x^16 + w*x^15 + x^14 + w^2*x^13 + w^2*x^12 + w*x^11 + w^2*x^10 + w^2*x^9 + x^8 + w*x^7 + w^2*x^6 + w*x^5 + w,  w*x^16 + w*x^15 + w^2*x^13 + w*x^12 + w^2*x^11 + x^10 + x^9 + x^8 + x^7 + w^2*x^6 + w*x^5 + w^2*x^4 + w*x^2 + w*x + w^2
[2]:  [[34, 23, 4]] quantum code over GF(2^2)
     Subcode of [1]

    stabilizer matrix:

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

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.