Bounds on the minimum distance of additive quantum codes

Bounds on [[59,29]]₂

lower bound:	7
upper bound:	10
Construction

Construction of a [[59,29,7]] quantum code:
[1]:  [[58, 30, 7]] quantum code over GF(2^2)
     QuasiCyclicCode of length 58 with generating polynomials: w^2*x^28 + x^27 + w^2*x^25 + w*x^24 + w*x^23 + x^22 + w*x^21 + w*x^19 + w*x^18 + x^15 + w*x^14 + x^13 + x^12 + w*x^10 + x^8 + x^7 + w*x^6 + w*x^5 + w*x^4 + 1,  x^28 + w*x^27 + w*x^26 + x^24 + w*x^23 + x^22 + w^2*x^21 + w*x^19 + w^2*x^18 + x^17 + x^15 + w^2*x^14 + w^2*x^13 + w*x^12 + w^2*x^11 + w*x^10 + w^2*x^9 + x^6 + w*x^3 + w^2*x^2 + w^2*x + w
[2]:  [[58, 29, 7]] quantum code over GF(2^2)
     Subcode of [1]
[3]:  [[59, 29, 7]] quantum code over GF(2^2)
     ExtendCode [2] by 1

    stabilizer matrix:

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

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 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 [[59,29]]2

Construction

Notes

Bounds on [[59,29]]₂
