Bounds on the minimum distance of additive quantum codes

Bounds on [[63,36]]₂

lower bound:	7
upper bound:	9
Construction

Construction type: GraQC
Construction of a [[63,36,7]] quantum code:
[1]:  [[63, 36, 7]] quantum code over GF(2^2)
     QuasiCyclicCode of length 63 stacked to height 2 with generating polynomials: w^2*x^62 + x^61 + w*x^59 + w*x^58 + w^2*x^55 + w^2*x^54 + w*x^53 + x^52 + x^50 + x^49 + w^2*x^47 + w^2*x^46 + w*x^44 + x^43 + w^2*x^42 + w*x^41 + x^40 + x^38 + x^36 + w^2*x^35 + w^2*x^34 + w*x^33 + w^2*x^32 + w*x^31 + w^2*x^30 + x^29 + x^28 + x^27 + x^26 + w^2*x^24 + w*x^22 + x^21 + w*x^20 + w*x^18 + w^2*x^15 + x^13 + w*x^11 + w*x^10 + w*x^9 + x^8 + w*x^7 + x^6 + w*x^5 + w^2*x^4 + w*x^3 + w^2*x^2 + w^2*x + w^2,  w^2*x^62 + w*x^61 + w*x^60 + w^2*x^59 + x^57 + w*x^55 + w^2*x^53 + x^52 + w^2*x^51 + w^2*x^50 + w*x^49 + w^2*x^48 + w^2*x^46 + w^2*x^45 + w*x^43 + x^41 + w^2*x^40 + w^2*x^36 + x^35 + x^34 + x^33 + w*x^32 + w^2*x^31 + w^2*x^30 + w^2*x^28 + x^27 + w*x^25 + w*x^24 + x^23 + w^2*x^21 + x^20 + w^2*x^19 + x^18 + w^2*x^17 + w*x^16 + w^2*x^15 + w^2*x^12 + w*x^11 + w*x^10 + w*x^9 + w*x^8 + x^6 + w*x^5 + w*x^4 + x^3 + w^2*x^2 + w^2*x + w

    stabilizer matrix:

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

last modified: 2020-09-17
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 [[63,36]]2

Construction

Notes

Bounds on [[63,36]]₂
