Bounds on the minimum distance of additive quantum codes

Bounds on [[27,9]]₂

Construction

Construction of a [[27,9,6]] quantum code:
[1]:  [[27, 9, 6]] quantum code over GF(2^2)
     QuasiCyclicCode of length 27 stacked to height 2 with generating polynomials: x^8 + 1,  w^2*x^8 + w*x^7 + x^6 + x^4 + x^3 + w*x,  w*x^8 + w*x^7 + x^6 + w*x^5 + w^2*x^3 + x^2 + x + w,  w*x^8 + w,  x^8 + w^2*x^7 + w*x^6 + w*x^4 + w*x^3 + w^2*x,  w^2*x^8 + w^2*x^7 + w*x^6 + w^2*x^5 + x^3 + w*x^2 + w*x + w^2

    stabilizer matrix:

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

last modified: 2006-04-04

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.