Bounds on the minimum distance of additive quantum codes

Bounds on [[68,40]]₂

lower bound:	7
upper bound:	9
Construction

Construction type: WangLiLvSong,
Construction of a [[68,40,7]] quantum code:
[1]:  [[68, 40, 7]] quantum code over GF(2^2)
     ConstructionX from linear cyclic codes
Construction from a stored generator matrix

    stabilizer matrix:

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

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.
This page is maintained by Markus Grassl (codes@codetables.de). Last change: 10.06.2024
Bounds on the minimum distance of additive quantum codes

Bounds on [[68,40]]2

Construction

Notes

Bounds on [[68,40]]₂
