Bounds on the minimum distance of additive quantum codes

Bounds on [[64,30]]₂

lower bound:	8
upper bound:	12
Construction

Construction of a [[64,30,8]] quantum code:
[1]:  [[62, 30, 8]] quantum code over GF(2^2)
     QuasiCyclicCode of length 62 stacked to height 2 with generating polynomials: w*x^30 + x^29 + w*x^28 + w^2*x^27 + w^2*x^26 + w*x^25 + w^2*x^20 + w*x^19 + x^18 + w^2*x^17 + x^16 + w*x^14 + w^2*x^11 + w^2*x^10 + w^2*x^9 + w^2*x^8 + x^7 + w^2*x^6 + w^2*x^5 + x^3 + w*x,  w*x^30 + x^29 + w^2*x^28 + w*x^27 + w^2*x^26 + w^2*x^25 + w^2*x^24 + w*x^23 + x^22 + w*x^21 + w*x^20 + w*x^19 + w^2*x^17 + x^16 + x^14 + w^2*x^13 + w*x^12 + w*x^10 + w*x^9 + w^2*x^8 + x^7 + w*x^6 + x^5 + x^4 + w*x^3 + x + w,  w^2*x^30 + w*x^29 + w^2*x^28 + x^27 + x^26 + w^2*x^25 + x^20 + w^2*x^19 + w*x^18 + x^17 + w*x^16 + w^2*x^14 + x^11 + x^10 + x^9 + x^8 + w*x^7 + x^6 + x^5 + w*x^3 + w^2*x,  w^2*x^30 + w*x^29 + x^28 + w^2*x^27 + x^26 + x^25 + x^24 + w^2*x^23 + w*x^22 + w^2*x^21 + w^2*x^20 + w^2*x^19 + x^17 + w*x^16 + w*x^14 + x^13 + w^2*x^12 + w^2*x^10 + w^2*x^9 + x^8 + w*x^7 + w^2*x^6 + w*x^5 + w*x^4 + w^2*x^3 + w*x + w^2
[2]:  [[64, 30, 8]] quantum code over GF(2^2)
     ExtendCode [1] by 2

    stabilizer matrix:

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

last modified: 2006-04-03
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.
This page is maintained by Markus Grassl (codes@codetables.de). Last change: 23.10.2014
Bounds on the minimum distance of additive quantum codes

Bounds on [[64,30]]2

Construction

Notes

Bounds on [[64,30]]₂
