Bounds on the minimum distance of additive quantum codes

Bounds on [[27,15]]₂

Construction

Construction of a [[27,15,4]] quantum code:
[1]:  [[40, 30, 4]] quantum code over GF(2^2)
     QuasiCyclicCode of length 40 stacked to height 2 with generating polynomials: 1,  w^2*x^4 + w*x^3 + w^2*x^2 + w*x + w^2,  x^4 + w^2*x^2 + w^2*x + w^2,  x^4 + w*x^3 + x^2,  w*x^4 + x^3 + w^2*x,  w*x^4 + x^3 + x^2 + x,  w^2*x^4 + x^2 + w*x,  x^4 + w^2*x^3 + w^2*x^2 + x + 1,  w,  x^4 + w^2*x^3 + x^2 + w^2*x + 1,  w*x^4 + x^2 + x + 1,  w*x^4 + w^2*x^3 + w*x^2,  w^2*x^4 + w*x^3 + x,  w^2*x^4 + w*x^3 + w*x^2 + w*x,  x^4 + w*x^2 + w^2*x,  w*x^4 + x^3 + x^2 + w*x + w
[2]:  [[24, 14, 4]] quantum code over GF(2^2)
     Shortening of [1] at { 3, 5, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 28, 30, 31, 34 }
[3]:  [[25, 14, 4]] quantum code over GF(2^2)
     ExtendCode [2] by 1
[4]:  [[27, 14, 4]] quantum code over GF(2^2)
     ExtendCode [3] by 2

    stabilizer matrix:

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

last modified: 2005-06-24

Further notes

The upper bound was shown in

Ching-Yi Lai and Alexei Ashikhmin,
"Linear Programming Bounds for Entanglement-Assisted Quantum Error-Correcting Codes by Split Weight Enmerators,"
IEEE Transactions on Information Theory, 64(1):622-639 (2018).
DOI: 10.1109/TIT.2017.2711601

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.