Bounds on the minimum distance of additive quantum codes

Bounds on [[67,52]]₂

Construction

Construction of a [[67,52,4]] quantum code:
[1]:  [[80, 66, 4]] quantum code over GF(2^2)
     QuasiCyclicCode of length 80 stacked to height 2 with generating polynomials: w*x^9 + x^7 + x^5 + x^4 + w*x^2 + x,  x^9 + w^2*x^8 + w*x^6 + w^2*x^4 + w^2*x^2,  w^2*x^9 + w^2*x^7 + w*x^6 + w^2*x + 1,  x^9 + w*x^8 + x^7 + w^2*x^6 + x^5 + w*x^3 + x + w^2,  w^2*x^9 + x^5 + x^4 + x^3 + x + w^2,  w^2*x^9 + w^2*x^8 + w^2*x^7 + w*x^6 + w*x^4 + w*x^3 + w*x + w^2,  w*x^9 + w^2*x^8 + w*x^7 + w^2*x^6 + x^5 + x^4 + x^3 + w*x^2 + x + w,  w^2*x^9 + w^2*x^8 + x^7 + x^6 + w*x^5 + w^2*x^3 + w*x^2 + w^2,  w^2*x^9 + w*x^7 + w*x^5 + w*x^4 + w^2*x^2 + w*x,  w*x^9 + x^8 + w^2*x^6 + x^4 + x^2,  x^9 + x^7 + w^2*x^6 + x + w,  w*x^9 + w^2*x^8 + w*x^7 + x^6 + w*x^5 + w^2*x^3 + w*x + 1,  x^9 + w*x^5 + w*x^4 + w*x^3 + w*x + 1,  x^9 + x^8 + x^7 + w^2*x^6 + w^2*x^4 + w^2*x^3 + w^2*x + 1,  w^2*x^9 + x^8 + w^2*x^7 + x^6 + w*x^5 + w*x^4 + w*x^3 + w^2*x^2 + w*x + w^2,  x^9 + x^8 + w*x^7 + w*x^6 + w^2*x^5 + x^3 + w^2*x^2 + 1
[2]:  [[66, 52, 4]] quantum code over GF(2^2)
     Shortening of [1] at { 14, 15, 19, 20, 31, 36, 52, 53, 57, 58, 63, 68, 74, 79 }
[3]:  [[67, 52, 4]] quantum code over GF(2^2)
     ExtendCode [2] by 1

    stabilizer matrix:

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

last modified: 2006-04-03

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.