Bounds on the minimum distance of additive quantum codes

Bounds on [[13,5]]₂

Construction

Construction of a [[13,5,3]] quantum code:
[1]:  [[11, 5, 3]] quantum code over GF(2^2)
     Construction from a stored generator matrix
[2]:  [[13, 5, 3]] quantum code over GF(2^2)
     ExtendCode [1] by 2

    stabilizer matrix:

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

last modified: 2011-06-23

Further notes

The upper bound was shown in

Jürgen Bierbrauer, Richard Fears, Stefano Marcugini, and Fernanda Pambianco,
"The Nonexistence of a [[13,5,4]]-Quantum Stabilizer Code,"
IEEE Transactions on Information Theory, 57(7):4788-4793 (2011).
DOI: 10.1109/TIT.2011.2146430

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 even 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.