Bounds on the minimum distance of additive quantum codes

Bounds on [[165,146]]₂

lower bound:	4
upper bound:	6
Construction

Construction of a [[165,146,4]] quantum code:
[1]:  [[252, 238, 4]] quantum code over GF(2^2)
     Construction from a stored generator matrix
[2]:  [[165, 151, 4]] quantum code over GF(2^2)
     Shortening of [1] at { 1, 2, 6, 10, 14, 18, 22, 25, 26, 29, 31, 32, 37, 41, 42, 43, 60, 63, 68, 72, 74, 79, 85, 86, 90, 91, 95, 96, 97, 102, 103, 111, 114, 115, 117, 119, 120, 121, 122, 123, 125, 126, 127, 128, 130, 134, 135, 136, 140, 143, 150, 151, 156, 159, 163, 165, 168, 170, 178, 181, 185, 188, 192, 193, 206, 208, 217, 221, 222, 224, 225, 226, 228, 231, 232, 233, 235, 236, 237, 238, 239, 242, 244, 246, 248, 249, 251 }
[3]:  [[165, 146, 4]] quantum code over GF(2^2)
     Subcode of [2]

    stabilizer matrix:

      [1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 1 1 0 0 1 0 1 0 0 1 1 0 1 0 0 1 0 1 0 0 1 1 1 1 1 0 1 1 1 1 0 1 1 0 1 0 0 0 0 1 0 0 1 0 1 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 1 0 1 1 0 0 0 1 1 0 1 1 0 1 0 0 0 0 1 0 1 0 1 0 1 1 1 0 1 0 1 1 0 1 1 1 0 0 0 1 1 1 0 0 1 0 0 1 0 0 1 1 1 1 1 0 1 0 1 0 0 0 0 1 1 1 0 1 1 1 1 1 0 1 1 1 1 0 1 0|0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 0 1 0 1 0 0 0 1 0 0 0 0 1 1 0 1 1 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 1 1 1 1 0 0 0 1 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 1 1 0 1 1 0 1 1 0 0 1 1 0 0 1 1 0 1 0 1 1 1 0 0 1 0 0]
      [0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 0 1 0 1 0 0 0 1 0 0 0 0 1 1 0 1 1 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 1 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 1 1 1 1 0 0 0 1 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 1 1 0 1 1 0 1 1 0 0 1 1 0 0 1 1 0 1 1 0 0 1 0 0 1 0 1|1 0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 1 0 1 1 0 1 0 1 1 0 1 1 0 1 0 0 1 1 0 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 1 1 1 1 1 0 0 1 1 1 0 1 1 1 0 0 0 1 0 0 0 1 1 0 0 1 1 0 0 0 1 1 1 0 0 0 1 1 0 1 0 0 1 1 0 0 1 1 1 1 1 0 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 1 1 0 1 1 0 1 1 1 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 1 1 0 1 1 0 1]
      [0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 1 0 0 0 1 1 0 0 1 1 0 0 1 0 1 1 0 1 1 1 0 1 0 1 1 0 0 0 1 1 0 0 1 1 0 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 0 0 1 1 1 0 1 1 0 1 0 1 0 0 1 1 0 0 1 1 0 1 1 1 0 0 1 0 0 1 0 1 0 0 1 0 1 1 1 0 0 1 1 0 1 0 1 0 1 1 1 0 1 1 1 1 0 1 1 1 1 0 0 0 1 0|0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 1 0 1 0 1 1 0 0 1 1 1 1 1 1 1 0 1 1 0 0 0 1 1 1 0 0 1 0 0 0 1 1 1 0 1 0 0 1 1 1 1 1 1 0 0 1 1 1 0 1 1 1 0 0 1 1 1 0 0 0 0 0 1 0 1 1 0 0 0 1 1 1 1 0 0 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 0 1 1 1 0 1 1 1 0 0 1 0 0 0 1 0 1 1 1 1 1 0 1 0 0 1 1 1 0 1 1 0 0 1 0 0 0 0 1 0 0 1 0 0]
      [0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 1 0 1 0 1 1 0 0 1 1 1 1 1 1 1 0 1 1 0 0 0 1 1 1 0 0 1 0 0 0 1 1 1 0 1 0 0 1 1 1 1 1 1 0 0 1 1 1 0 1 1 1 0 0 1 1 1 0 0 0 0 0 1 0 1 1 0 0 0 1 1 1 1 0 0 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 0 1 1 1 1 1 0 1 1 1 0 1 1 1 0 0 1 0 0 0 1 0 1 1 1 1 1 0 1 0 0 1 1 1 0 1 1 0 0 1 0 0 1 1 1 1 1 1 1 1|0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 0 0 1 1 0 1 0 1 0 1 0 0 1 1 0 1 1 0 1 1 1 1 1 0 1 1 1 1 0 0 1 0 1 1 1 0 0 0 0 1 0 0 1 0 1 0 0 1 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1 1 0 0 1 0 1 1 0 0 0 0 1 0 1 1 0 1 0 1 0 0 1 0 1 0 1 1 1 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 0 0 0 0 1 1 0 0 1 1 0 1 1 0 0 0 0 0 0 0 1 0 1 1]
      [0 0 1 0 0 0 0 0 0 1 0 1 0 1 1 0 1 0 1 1 1 0 1 1 0 0 1 1 1 0 1 0 0 1 1 0 0 1 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 0 0 1 0 0 1 0 1 0 1 0 0 1 1 0 0 1 0 1 0 0 0 1 0 0 1 1 0 0 1 1 0 1 1 0 1 0 1 1 0 1 0 1 1 0 0 0 0 1 1 1 1 0 1 1 0 1 0 0 1 0 1 0 0 1 1 0 1 0 0 1 1 0 0 0 0 1 1 0 1 1 1 1 0 0 1 0 1 1 1 0 1 0 1 0 0 0 1 1 0 1 1 0 0 1 1 0 0 1 1 0 0|0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 1 1 1 1 1 1 0 1 0 0 1 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 0 1 1 1 1 1 1 0 0 1 1 0 0 1 0 1 1 1 0 1 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 1 1 1 0 0 1 0 1 0 1 1 0 0 0 0 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 1 0 1 0 1 1 1 1 0 1 0 0 1 0 0 1 0 0 1 1 1 1 0 1 0 0 1 0 1 1 0 1 0]
      [0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 1 1 1 1 1 1 0 1 0 0 1 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 0 1 1 1 1 1 1 0 0 1 1 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 1 1 1 0 0 1 0 1 0 1 1 0 0 0 0 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 1 0 1 0 1 1 1 1 0 1 0 0 1 0 0 1 0 0 1 1 1 1 0 1 0 1 0 0 0 1 0 0 0|0 0 1 0 0 0 0 0 0 1 0 1 1 1 0 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 1 0 1 1 1 1 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 0 1 1 0 0 1 0 1 1 0 0 0 1 1 0 1 1 0 1 1 1 0 1 1 0 1 1 1 1 0 1 1 0 1 0 0 1 0 1 1 1 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 1 0 1 1 0 1 0 1 1 0 0 1 0 1 1 1 0 1 1 0 1 0 0 0 1 1 0 0 0 0 1 0 1 1 0 0 1 0 0 1 1 1 0]
      [0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 0 0 0 0 1 0 1 1 0 0 1 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 1 1 0 0 1 1 0 1 0 1 1 0 1 1 1 0 1 0 1 0 1 0 0 0 1 1 0 0 0 1 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 0 1 1 0 0 0 1 0 0 1 1 0 0 0 1 0 1 1 0 1 1 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 0 1 0 1 0 1 0|0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 0 0 1 1 0 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 0 1 1 0 1 1 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 1 0 1 1 0 1 1 0 0 0 1 1 0 1 1 1 0 1 1 1 1 0]
      [0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 0 0 1 1 0 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 0 1 1 0 1 1 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 1 0 1 1 0 1 1 0 0 0 1 1 0 1 1 1 0 1 1 1 1 0|0 0 0 1 0 0 0 0 0 0 0 0 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 0 1 1 0 1 1 0 0 1 1 1 0 0 1 1 0 1 1 1 0 1 1 1 1 0 1 1 1 1 1 0 1 0 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 1 1 1 1 1 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 1 1 0 1 1 0 0 1 0 1 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 0 1 0 0]
      [0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 1 1 1|0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 0 1 1 1 1 0 0 0 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 0 1 1 1 0 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 1 1 1 1 0 1 0 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 1 1 1 0 1 0 1 1 1 1 0 0 0 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 0 1 1 1 0 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 1 1 1 1 1 0 1 0 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1|0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 1 1]
      [0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 1 0 1 1 1 1 1 0 1 0 1 1 1 1 1 0 0 1 0 1 1 1 0 1 0 1 1 1 0 1 1 0 0 0 0 1 0 1 0 1 1 1 1 1 1 1 0 1 0 0 1 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 1 0 1 1 1 1 0 1 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 0 0 0 1 1 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 1 0 1 1 0 1 0 0|0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 1 1 0 0 1 0 0 1 1 0 1 1 1 1 1 0 1 1 1 1 1 1 0 1 0 0 0 1 1 1 1 0 1 1 1 0 1 0 1 1 1 1 1 0 0 0 0 1 0 1 0 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 1 0 1 1 0 1 0 1 0 1 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 1 0 1 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 1 0 0]
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last modified: 2009-02-08
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 [[165,146]]2

Construction

Notes

Bounds on [[165,146]]₂
