Bounds on the minimum distance of additive quantum codes

Bounds on [[67,39]]₂

lower bound:	7
upper bound:	9
Construction

Construction type: EzermanGrasslLingOzbudakOzkaya
Construction of a [[67,39,7]] quantum code:
[1]:  [65, 14] Quasicyclic of degree 5 Linear Code over GF(2^2)
     QuasiCyclicCode of length 65 stacked to height  2 with generating polynomials: 1,  w*x^11 + w^2*x^10 + w*x^9 + w^2*x^8 + w^2*x^7 + w^2*x^5 + w^2*x^4 + w^2*x + w,  x^9 + x^8 + x^6 + x^5 + x^4 + w^2*x^2 + w,  x^12 + x^11 + w*x^10 + w^2*x^8 + w^2*x^7 + x^5 + w*x^3 + w^2*x^2 + w^2*x,  w*x^12 + x^11 + w^2*x^10 + w*x^9 + w*x^8 + w*x^6 + x^4 + w*x^3 + x + w^2,  0,  x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1,  w^2*x^12 + w^2*x^11 + w^2*x^10 + w^2*x^9 + w^2*x^8 + w^2*x^7 + w^2*x^6 + w^2*x^5 + w^2*x^4 + w^2*x^3 + w^2*x^2 + w^2*x + w^2,  w^2*x^12 + w^2*x^11 + w^2*x^10 + w^2*x^9 + w^2*x^8 + w^2*x^7 + w^2*x^6 + w^2*x^5 + w^2*x^4 + w^2*x^3 + w^2*x^2 + w^2*x + w^2,  w^2*x^12 + w^2*x^11 + w^2*x^10 + w^2*x^9 + w^2*x^8 + w^2*x^7 + w^2*x^6 + w^2*x^5 + w^2*x^4 + w^2*x^3 + w^2*x^2 + w^2*x + w^2
[2]:  [[67, 39, 7]] quantum code over GF(2^2)
     QuantumConstructionX applied to [1] with e = 2

    stabilizer matrix:

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

last modified: 2024-06-15
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.
This page is maintained by Markus Grassl (codes@codetables.de). Last change: 10.06.2024
Bounds on the minimum distance of additive quantum codes

Bounds on [[67,39]]2

Construction

Notes

Bounds on [[67,39]]₂
