Bounds on the minimum distance of additive quantum codes

Bounds on [[123,109]]₂

lower bound:	4
upper bound:	4
Construction

Construction of a [[123,109,4]] quantum code:
[1]:  [[252, 238, 4]] quantum code over GF(2^2)
     Construction from a stored generator matrix
[2]:  [[123, 109, 4]] quantum code over GF(2^2)
     Shortening of [1] at { 1, 2, 6, 8, 10, 12, 13, 15, 17, 19, 20, 23, 27, 29, 32, 33, 34, 37, 38, 39, 40, 41, 42, 45, 46, 50, 51, 53, 54, 56, 57, 58, 61, 68, 70, 73, 74, 75, 77, 78, 80, 82, 83, 86, 88, 90, 91, 93, 94, 95, 96, 98, 99, 103, 104, 111, 112, 113, 115, 116, 119, 120, 121, 123, 129, 130, 131, 132, 133, 134, 136, 138, 139, 140, 143, 144, 146, 147, 151, 152, 153, 154, 155, 162, 163, 165, 166, 168, 169, 170, 171, 175, 176, 178, 183, 185, 187, 189, 190, 191, 193, 194, 197, 200, 203, 204, 206, 209, 211, 215, 217, 219, 220, 221, 224, 225, 227, 228, 229, 230, 231, 232, 236, 237, 242, 244, 245, 246, 248 }

    stabilizer matrix:

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

last modified: 2006-04-03
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.
This page is maintained by Markus Grassl (codes@codetables.de). Last change: 23.10.2014
Bounds on the minimum distance of additive quantum codes

Bounds on [[123,109]]2

Construction

Notes

Bounds on [[123,109]]₂
