Bounds on the minimum distance of additive quantum codes

Bounds on [[64,53]]₂

Construction

Construction of a [[64,53,4]] quantum code:
[1]:  [[64, 53, 4]] quantum code over GF(2^2)
     Quantum Twisted Code of length 64 with interval [ 27, 38, 52 ] and parameter kappa 3

    stabilizer matrix:

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

last modified: 2024-04-24

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.