Bounds on the minimum distance of additive quantum codes

Bounds on [[16,6]]₃

Construction

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

    stabilizer matrix:

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

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.