Bounds on the minimum distance of additive quantum codes

Bounds on [[52,16]]₂

lower bound:	10
upper bound:	13
Construction

Construction type: DastbastehShivj
Construction of a [[52,16,10]] quantum code:
[1]:  [51, 18 : 36, 17] GF(2)-additive Code over GF(2^2)
     additive QuasiCyclicCode of length 51 stacked to height 2 with generating polynomials: x^50 + x^49 + w^2*x^48 + w^2*x^46 + w*x^45 + w^2*x^44 + w^2*x^42 + w^2*x^41 + w*x^40 + w^2*x^39 + w^2*x^38 + w^2*x^37 + x^35 + w*x^32 + x^31 + w^2*x^29 + x^26 + w^2*x^25 + w^2*x^24 + x^23 + x^22 + x^21 + x^20 + w^2*x^19 + x^18 + x^17 + 1,  x^50 + w^2*x^49 + w^2*x^48 + w^2*x^47 + w*x^46 + w^2*x^45 + w*x^44 + w^2*x^43 + w^2*x^41 + w*x^40 + x^39 + w^2*x^38 + w*x^37 + x^36 + x^35 + w*x^34 + w*x^33 + w*x^32 + x^31 + x^29 + w*x^28 + x^27 + w*x^25 + w*x^24 + x^23 + w^2*x^21 + w*x^20 + x^19 + x^18 + x^17 + w
[2]:  [[52, 16, 10]] quantum code over GF(2^2)
     QuantumConstructionX applied to [1] with e = 1

    stabilizer matrix:

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

last modified: 2024-05-06
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 [[52,16]]2

Construction

Notes

Bounds on [[52,16]]₂
