Bounds on the minimum distance of additive quantum codes

Bounds on [[23,12]]₃

Construction

Construction of a [[23,12,5]] quantum code:
[1]:  [[23, 12, 5]] quantum code over GF(3^2)
     cyclic code of length 23 with generating polynomial x^19 + w^7*x^18 + w^7*x^17 + w^3*x^16 + 2*x^15 + w^2*x^14 + w^7*x^13 + 2*x^12 + w^6*x^11 + 2*x^10 + w^7*x^9 + w^7*x^8 + x^7 + w*x^5 + x^4 + w*x^2 + w^5*x + w^2

    stabilizer matrix:

      [1 0 0 0 0 0 2 2 0 0 0 0 2 1 2 2 2 2 2 2 1 0 0|0 0 0 0 0 2 2 0 2 0 0 1 1 1 1 0 1 0 2 2 2 0 1]
      [0 0 0 0 0 0 0 2 0 2 2 0 1 2 0 1 2 2 2 2 1 2 0|1 0 0 0 0 1 1 1 2 1 2 1 1 2 1 2 0 0 2 1 0 0 2]
      [0 1 0 0 0 0 0 0 2 1 1 0 2 0 1 1 0 0 0 0 1 2 0|0 0 0 0 0 2 1 1 1 1 1 2 0 2 0 2 0 1 1 1 2 2 1]
      [0 0 0 0 0 0 0 2 2 2 1 2 1 0 2 1 0 1 1 1 0 0 2|0 1 0 0 0 1 2 2 0 0 0 0 2 0 0 0 2 0 2 0 1 0 2]
      [0 0 1 0 0 0 0 1 0 0 2 1 2 0 0 0 2 1 1 1 2 2 2|0 0 0 0 0 2 1 0 2 0 2 0 1 1 1 1 2 0 2 0 1 2 0]
      [0 0 0 0 0 0 2 1 2 1 1 1 2 1 2 2 2 1 2 2 0 2 0|0 0 1 0 0 0 1 0 0 1 2 2 2 2 2 2 1 2 1 2 2 1 0]
      [0 0 0 1 0 0 2 2 1 0 0 2 0 0 2 2 2 1 0 0 2 2 2|0 0 0 0 0 2 1 1 2 2 0 0 1 2 2 1 2 2 2 1 2 1 0]
      [0 0 0 0 0 0 0 2 1 2 1 1 1 2 1 2 2 2 1 2 2 0 2|0 0 0 1 0 0 0 1 0 0 1 2 2 2 2 2 2 1 2 1 2 2 1]
      [0 0 0 0 1 0 2 1 2 1 0 0 1 1 2 1 1 1 0 2 1 2 2|0 0 0 0 0 2 1 1 0 2 2 1 1 2 0 2 2 2 1 1 0 2 2]
      [0 0 0 0 0 0 2 0 2 2 0 1 2 0 1 2 2 2 2 1 2 0 0|0 0 0 0 1 1 1 2 1 2 1 1 2 1 2 0 0 2 1 0 0 2 1]
      [0 0 0 0 0 1 2 0 1 1 0 0 0 1 0 2 2 2 2 1 1 0 2|0 0 0 0 0 0 2 2 2 1 1 1 0 1 1 2 0 2 0 1 0 0 2]

last modified: 2024-06-10

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.