Bounds on the minimum distance of additive quantum codes

Bounds on [[83,73]]₂

Construction

Construction of a [[83,73,3]] quantum code:
[1]:  [[93, 83, 3]] quantum code over GF(2^2)
     QuasiCyclicCode of length 93 stacked to height 2 with generating polynomials: w*x^30 + w^2*x^29 + w^2*x^27 + x^25 + x^24 + w*x^23 + w*x^20 + x^19 + w^2*x^18 + x^17 + w*x^16 + w*x^15 + x^14 + x^13 + x^11 + w^2*x^10 + w^2*x^9 + w^2*x^8 + x^7 + w^2*x^5 + w*x^4 + w*x^2 + w^2*x + w,  x^30 + w^2*x^29 + x^28 + x^27 + w^2*x^26 + w^2*x^24 + w*x^22 + w*x^21 + x^20 + x^17 + w*x^16 + w^2*x^15 + w*x^14 + x^13 + x^12 + w*x^11 + w*x^10 + w*x^8 + w^2*x^7 + w^2*x^6 + w^2*x^5 + w*x^4 + w^2*x^2 + x,  w^2*x^29 + w^2*x^28 + w*x^27 + w*x^26 + w*x^25 + w^2*x^23 + x^22 + x^21 + w*x^19 + x^18 + x^17 + w*x^16 + x^15 + w^2*x^14 + x^13 + w^2*x^12 + w^2*x^11 + x^9 + w*x^6 + w^2*x^5 + w*x^4 + x^2 + w*x + w^2,  w^2*x^30 + x^29 + x^27 + w*x^25 + w*x^24 + w^2*x^23 + w^2*x^20 + w*x^19 + x^18 + w*x^17 + w^2*x^16 + w^2*x^15 + w*x^14 + w*x^13 + w*x^11 + x^10 + x^9 + x^8 + w*x^7 + x^5 + w^2*x^4 + w^2*x^2 + x + w^2,  w*x^30 + x^29 + w*x^28 + w*x^27 + x^26 + x^24 + w^2*x^22 + w^2*x^21 + w*x^20 + w*x^17 + w^2*x^16 + x^15 + w^2*x^14 + w*x^13 + w*x^12 + w^2*x^11 + w^2*x^10 + w^2*x^8 + x^7 + x^6 + x^5 + w^2*x^4 + x^2 + w*x,  x^29 + x^28 + w^2*x^27 + w^2*x^26 + w^2*x^25 + x^23 + w*x^22 + w*x^21 + w^2*x^19 + w*x^18 + w*x^17 + w^2*x^16 + w*x^15 + x^14 + w*x^13 + x^12 + x^11 + w*x^9 + w^2*x^6 + x^5 + w^2*x^4 + w*x^2 + w^2*x + 1
[2]:  [[83, 73, 3]] quantum code over GF(2^2)
     Shortening of [1] at { 17, 25, 59, 60, 62, 80, 82, 83, 87, 91 }

    stabilizer matrix:

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