Bounds on the minimum distance of additive quantum codes

Bounds on [[28,14]]₂

Construction

Construction of a [[28,14,5]] quantum code:
[1]:  [[28, 14, 5]] quantum code over GF(2^2)
     Construction from a stored generator matrix

    stabilizer matrix:

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

last modified: 2020-09-18

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.