Bounds on the minimum distance of additive quantum codes

Bounds on [[240,190]]₂

Construction

Construction of a [[240,190,8]] quantum code:
[1]:  [234, 24] Quasicyclic of degree 18 Linear Code over GF(2^2)
     QuasiCyclicCode of length 234 stacked to height  6 with generating polynomials: 1,  w*x^5 + x^3 + w^2*x^2 + w*x,  w*x^9 + w*x^8 + w^2*x^5 + w*x^3 + 1,  w^2*x^11 + x^9 + w^2*x^8 + x^7 + w*x^6 + x^5 + x^4 + w*x^3 + w*x^2 + w^2*x + 1,  w*x^11 + w*x^10 + x^8 + w*x^7 + w*x^6 + w^2*x^3 + w^2*x + w,  x^11 + w^2*x^10 + w^2*x^9 + w^2*x^8 + w*x^7 + x^6 + x^4 + w*x^3 + w*x^2 + w^2*x + 1,  w^2*x^12 + x^10 + w*x^9 + w*x^8 + x^7 + x^2 + w,  x^12 + w*x^11 + w*x^9 + x^8 + w*x^5 + w*x^4 + w^2*x^3 + x^2 + w^2,  w^2*x^12 + w*x^11 + w*x^10 + w*x^9 + w^2*x^8 + w*x^7 + x^6 + w*x^5 + x^3 + w^2*x^2 + w^2*x + w^2,  w*x^12 + w^2*x^11 + x^9 + w^2*x^7 + w^2*x^6 + w*x^5 + w*x^2 + w*x,  x^12 + w*x^11 + w*x^10 + x^7 + x^5 + w^2*x^4 + x^3 + w*x^2 + 1,  x^11 + w^2*x^8 + w*x^7 + w^2*x^5 + w*x^4 + w^2*x^3 + w^2*x^2 + x,  w^2*x^12 + x^10 + w*x^8 + x^7 + w^2*x^6 + w*x^4 + x^3 + x,  w^2*x^12 + x^11 + w*x^10 + w*x^9 + w*x^8 + w^2*x^7 + x^6 + x^4 + w^2*x^3 + w*x^2 + w*x + w,  w^2*x^12 + w*x^11 + w^2*x^10 + w^2*x^7 + w*x^6 + w*x^5 + w^2*x^2 + x + 1,  x^11 + w^2*x^10 + w^2*x^9 + x^8 + x^7 + x^5 + w*x^4 + w*x^3 + w^2*x^2 + w^2*x + 1,  w*x^10 + w^2*x^8 + x^6 + w*x^5 + x^2 + w^2*x + w,  x^8 + w*x^7 + x^6 + x^5 + w*x^4 + w^2*x^3 + x + 1,  0,  x^6 + w*x^5 + w^2*x^3 + w*x + 1,  w*x^11 + w^2*x^10 + w^2*x^9 + x^8 + w*x^7 + w^2*x^6 + w*x^5 + x^4 + x^2 + w^2*x + w,  x^11 + w^2*x^10 + w^2*x^9 + w^2*x^8 + x^7 + w^2*x^5 + w^2,  w^2*x^11 + x^10 + w^2*x^7 + w*x^2 + w*x + 1,  x^11 + w^2*x^10 + x^9 + w^2*x^8 + w^2*x^7 + x^5 + x^4 + w^2*x^3 + w*x^2 + w^2*x + w^2,  w^2*x^12 + w*x^11 + x^10 + w*x^8 + w^2*x^7 + w*x^6 + x^5 + x^4 + w^2*x^3 + w*x^2 + w^2*x,  x^12 + x^11 + w*x^10 + x^9 + x^6 + w*x^5 + x^4 + x^3 + x^2 + w^2*x + 1,  w^2*x^12 + x^11 + w^2*x^10 + x^8 + w*x^7 + w^2*x^6 + x^5 + x^4 + w^2*x^3 + w*x^2 + x,  x^11 + w*x^10 + x^9 + x^7 + w*x^6 + x^5 + w*x^3 + w,  w*x^12 + x^11 + w^2*x^10 + w*x^9 + w*x^8 + x^7 + x^5 + w*x^4 + w^2*x^3 + w^2*x + 1,  w^2*x^12 + w^2*x^11 + x^10 + w*x^9 + w^2*x^7 + x^4 + x^3 + x^2 + w*x + w^2,  w^2*x^12 + x^11 + w^2*x^10 + w^2*x^9 + w^2*x^8 + x^6 + x^5 + w*x^4 + w*x^3 + 1,  w^2*x^12 + w*x^11 + w*x^10 + w*x^9 + w^2*x^8 + x^7 + x^6 + w*x^5 + w^2*x^4 + w^2*x^3 + w*x^2 + x + 1,  w^2*x^12 + x^11 + w^2*x^10 + w^2*x^9 + w*x^8 + x^7 + w^2*x^6 + w^2*x^5 + w*x^3 + w*x + w,  w*x^11 + w*x^10 + x^9 + x^8 + w*x^7 + w*x^6 + w*x^4 + w,  w^2*x^10 + x^8 + w^2*x^7 + w*x^6 + x^5 + x^4 + w^2*x^3 + w^2*x^2 + w*x + 1,  w^2*x^12 + w^2*x^11 + x^10 + w^2*x^9 + w*x^8 + w*x^7 + x^6 + w^2*x^2 + w^2*x + w^2,  0,  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,  0,  0,  0,  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,  0,  0,  w*x^12 + w*x^11 + w*x^10 + w*x^9 + w*x^8 + w*x^7 + w*x^6 + w*x^5 + w*x^4 + w*x^3 + w*x^2 + w*x + w,  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,  x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1,  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*x^12 + w*x^11 + w*x^10 + w*x^9 + w*x^8 + w*x^7 + w*x^6 + w*x^5 + w*x^4 + w*x^3 + w*x^2 + w*x + w,  0,  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,  0,  w*x^12 + w*x^11 + w*x^10 + w*x^9 + w*x^8 + w*x^7 + w*x^6 + w*x^5 + w*x^4 + w*x^3 + w*x^2 + w*x + w,  0,  0,  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,  0,  0,  w*x^12 + w*x^11 + w*x^10 + w*x^9 + w*x^8 + w*x^7 + w*x^6 + w*x^5 + w*x^4 + w*x^3 + w*x^2 + w*x + w,  0,  w*x^12 + w*x^11 + w*x^10 + w*x^9 + w*x^8 + w*x^7 + w*x^6 + w*x^5 + w*x^4 + w*x^3 + w*x^2 + w*x + w,  0,  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,  0,  w*x^12 + w*x^11 + w*x^10 + w*x^9 + w*x^8 + w*x^7 + w*x^6 + w*x^5 + w*x^4 + w*x^3 + w*x^2 + w*x + w,  x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1,  0,  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,  0,  0,  0,  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,  0,  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*x^12 + w*x^11 + w*x^10 + w*x^9 + w*x^8 + w*x^7 + w*x^6 + w*x^5 + w*x^4 + w*x^3 + w*x^2 + w*x + w,  x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1,  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*x^12 + w*x^11 + w*x^10 + w*x^9 + w*x^8 + w*x^7 + w*x^6 + w*x^5 + w*x^4 + w*x^3 + w*x^2 + w*x + w,  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*x^12 + w*x^11 + w*x^10 + w*x^9 + w*x^8 + w*x^7 + w*x^6 + w*x^5 + w*x^4 + w*x^3 + w*x^2 + w*x + w,  w*x^12 + w*x^11 + w*x^10 + w*x^9 + w*x^8 + w*x^7 + w*x^6 + w*x^5 + w*x^4 + w*x^3 + w*x^2 + w*x + w,  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,  0,  w*x^12 + w*x^11 + w*x^10 + w*x^9 + w*x^8 + w*x^7 + w*x^6 + w*x^5 + w*x^4 + w*x^3 + w*x^2 + w*x + w,  0,  0,  0,  0,  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,  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,  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*x^12 + w*x^11 + w*x^10 + w*x^9 + w*x^8 + w*x^7 + w*x^6 + w*x^5 + w*x^4 + w*x^3 + w*x^2 + w*x + w,  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,  x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1,  0,  w*x^12 + w*x^11 + w*x^10 + w*x^9 + w*x^8 + w*x^7 + w*x^6 + w*x^5 + w*x^4 + w*x^3 + w*x^2 + w*x + w,  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
[2]:  [[238, 190, 8]] quantum code over GF(2^2)
     QuantumConstructionX applied to [1] with e = 4
[3]:  [[240, 190, 8]] quantum code over GF(2^2)
     ExtendCode [2] by 2

    stabilizer matrix:

      [1 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 1 0 1 1 0 1 1 0 0 0 0 0 0 1 1 0 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0 1 0 1 0 0 1 0 1 1 1 0 1 0 1 1 0 1 0 1 0 1 0 0 0 1 0 0 0 0 1 0 1 1 0 0 1 0 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 0 0 1 1 0 1 1 1 0 1 0 1 1 1 1 0 0 0 1 0 1 0 0 1 1 0 0 1 0 0 0 1 0 0 1 0 0 1 1 0 1 1 1 0 0 0 0 1 1 1 1 1 1 1 0 0 1 0 0 0 1 0 0 1 0 1 1 0 0 0 0 0 1 0 0 0 1 1 0 1 1 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 0 0 0 0 0 1 0 1 1 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 1 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 0 1 1 0 1 1 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 1 0 0 0 1 0 0 1 0 1 1 1 0 1 0 0 0 0 1 0 1 0 0 0 1 1 0 0 1 0 1 0 1 1 0 1 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 1 1 1 1 0 0 1 0 1 1 0 1 0 1 0 0 0 1 1 1 0 1 0 1 1 0 1 0 0 1 1 1 1 0 1 0 1 1 1 0 0 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 0 0 0 1 0 1 0 0 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 1 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 0 0 1 1 0 1 1 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 1 0 0 0 1 0 0 1 0 1 1 1 0 1 0 0 0 0 1 0 1 0 0 0 1 1 0 0 1 0 1 0 1 1 0 1 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 0 0 1 1 1 1 0 0 1 0 1 1 0 1 0 1 0 0 0 1 1 1 0 1 0 1 1 0 1 0 0 1 1 1 1 0 1 0 1 1 1 0 0 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 1 0 1 1 0 1 1 0 0 0 0 0 0 0 0 1 0 1 0 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 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 1 1 0 0 1 1 1 1 0 0 1 1 0 1 1 0 0 1 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 1 1 1 1 1 1 0 1 0 1 0 0 1 0 1 1 1 0 0 1 0 1 1 1 1 1 0 0 1 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 0 1 0 0 1 0 1 1 1 1 0 0 0 0 0 1 0 1 1 1 0 1 0 0 1 1 0 1 0 1 0 0 1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 0 0 1 0 0 1 1 1 0 0 1 0 1 0 0 1 0 0 0 0 1 1 1 1 1 1 0 1 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 1 0 0 0 0 1 0 1 1 0 1 0 1 0 1 0 1 0 1 0 0 1 1 0 1 1 1 1 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 1 0 0 1 1 1 1 0 0 0 0 0 1 1 0 1 1 1 0 0 1 0 1 0 0 0 0 1 1 1 0 0 1 1 0 1 0 1 0 1 1 1 1 0 1 1 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1 1 1 0 1 1 0 0 0 0 1 1 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1 0 0 1 0 0 1 1 1 0 1 0 1 1 1 0 1 0 1 1 1 0 1 1 0 1 1 0 1 0 0 0 0 1 1 0 1 1 0 1 0 1 0 0 0 1 1 0 0 0 1 1 0 0 0 1 1 1 0 1 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 1 1 0 0 1 0 0 0 0 0 0 1 1 1 0 0 1 1 1 1 1 1 1 0 1 1 1 1 0 0 1 0 0 1 1 0 0 1 1 1 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 1 0 1 0 0 0 1 1 0 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 1 0 0 1 1 1 0 1 0 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 1 1 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 1 0 1 0 1 1 0 0 1 1 1 1 1 0 0 1 1 1 1 0 0 0 1 0 0 1 1 1 1 0 0 1 0 1 1 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 1 1 0 0 1 0 0 0 0 0 0 1 1 1 0 0 1 1 1 1 1 1 1 0 1 1 1 1 0 0 1 0 0 1 1 0 0 1 1 1 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 1 0 1 0 0 0 1 1 0 1 1 0 1 1 1 0 1 0 0 1 0 0 1 0 1 0 0 1 1 1 0 1 0 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 1 1 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0 1 0 1 1 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 1 1 1 1 0 1 0 1 1 0 0 1 0 1 0 1 1 0 0 1 1 1 1 1 0 0 1 1 1 1 0 0 0 1 0 0 1 1 1 1 0 0 1 0 1 1 1 0 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 1 0 0 0 1 0 0 1 0 0 1 0 1 0 1 1 0 1 1 0 1 0 0 1 0 0 0 1 1 1 1 1 0 0 1 1 0 1 0 1 0 0 0 1 1 0 0 1 1 1 1 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 1 1 1 1 0 0 1 1 0 0 0 1 0 0 1 1 0 0 1 0 0 1 0 1 1 0 1 1 1 0 1 0 1 0 1 1 1 1 1 1 0 0 0 1 0 0 1 0 1 1 1 1 0 0 0 0 0 1 0 1 1 0 0 1 0 1 1 1 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 0 0 1 0 0 1 1 1 1 0 1 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 1 1 1 1 0 0]
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last modified: 2024-05-23

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.