Bounds on the minimum distance of additive quantum codes

Bounds on [[48,11]]₂

lower bound:	9
upper bound:	14
Construction

Construction of a [[48,11,9]] quantum code:
[1]:  [[45, 11, 9]] quantum code over GF(2^2)
     QuasiCyclicCode of length 45 stacked to height  2 with generating polynomials: x^4 + x^3 + w*x^2 + w,  w^2*x^8 + x^6 + w^2*x^5 + w^2*x^4 + w^2*x^3 + w^2*x^2 + w^2*x + 1,  w*x^13 + x^12 + w*x^9 + w^2*x^8 + w^2*x^5 + x^4 + x^2 + x,  0,  x^9 + x^8 + w*x^7 + w*x^6 + w*x^5 + x^4 + w^2*x + w,  x^14 + x^13 + x^12 + x^11 + w*x^10 + w*x^8 + w^2*x^7 + w^2*x^6 + x^5 + w^2*x^4 + x^3 + w^2*x^2 + x + w
[2]:  [[48, 11, 9]] quantum code over GF(2^2)
     ExtendCode [1] by 3

    stabilizer matrix:

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

last modified: 2024-06-15
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 [[48,11]]2

Construction

Notes

Bounds on [[48,11]]₂
