Bounds on the minimum distance of additive quantum codes

Bounds on [[55,20]]₂

lower bound:	9
upper bound:	13
Construction

Construction type: GuanLiLuYao
Construction of a [[55,20,9]] quantum code:
[1]:  [[55, 20, 9]] quantum code over GF(2^2)
     cyclic code of length 55 with generating polynomial x^54 + w*x^52 + w^2*x^51 + x^50 + w*x^48 + w*x^46 + w^2*x^45 + w^2*x^44 + x^43 + w^2*x^42 + w^2*x^40 + x^39 + w*x^37 + w^2*x^36 + w^2*x^35 + x^34 + x^33 + w^2*x^32 + w*x^30 + x^29 + x^28 + w*x^27 + x^26 + w*x^25 + x^24 + w*x^23 + x^22 + x^21 + w*x^8

    stabilizer matrix:

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

last modified: 2022-08-02
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 [[55,20]]2

Construction

Notes

Bounds on [[55,20]]₂
