Bounds on the minimum distance of additive quantum codes

Bounds on [[59,21]]₂

lower bound:	9
upper bound:	14
Construction

Construction type: EzermanGrasslLingOzbudakOzkaya
Construction of a [[59,21,9]] quantum code:
[1]:  [55, 19 : 38] GF(2)-additive Code over GF(2^2)
     additive QuasiCyclicCode of length 55 stacked to height 2 with generating polynomials: x^54 + x^53 + w*x^52 + w^2*x^50 + x^49 + x^48 + x^47 + x^46 + x^45 + x^44 + w^2*x^41 + x^40 + w^2*x^39 + w*x^37 + w*x^36 + w*x^35 + w^2*x^34 + w^2*x^33 + w*x^32 + w^2*x^31 + w*x^29 + w^2*x^28 + w*x^27 + w*x^26 + w^2*x^25 + w*x^24 + x^20 + w*x^18 + w^2*x^17 + w^2*x^16 + w*x^15 + w^2*x^14 + w*x^13 + w*x^12 + w*x^11 + x^10 + x^9 + w*x^8 + x^7 + w*x^5 + x^4 + w^2*x^3 + w*x^2 + w^2*x + 1,  x^54 + w*x^52 + x^51 + x^49 + x^48 + w^2*x^45 + x^44 + x^42 + x^41 + w^2*x^40 + w^2*x^39 + w^2*x^38 + w^2*x^36 + x^35 + w*x^34 + w*x^33 + w^2*x^32 + w^2*x^31 + w^2*x^30 + x^29 + w^2*x^28 + w^2*x^27 + x^26 + x^23 + w^2*x^21 + w^2*x^19 + x^18 + w^2*x^17 + x^16 + x^15 + x^13 + x^11 + x^8 + w*x^7 + w*x^6 + w*x^5 + w*x^3 + w*x^2 + w
[2]:  [[59, 21, 9]] quantum code over GF(2^2)
     QuantumConstructionX applied to [1] with e = 4

    stabilizer matrix:

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

last modified: 2024-11-17
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 [[59,21]]2

Construction

Notes

Bounds on [[59,21]]₂
