Bounds on the minimum distance of additive quantum codes

Bounds on [[36,27]]₂

Construction

Construction of a [[36,27,3]] quantum code:
[1]:  [[35, 28, 3]] quantum code over GF(2^2)
     cyclic code of length 35 with generating polynomial w^2*x^31 + x^29 + w*x^28 + w^2*x^27 + x^26 + w*x^24 + w^2*x^23 + w*x^22 + w^2*x^21 + w^2*x^20 + w^2*x^18 + x^17 + w*x^16 + w*x^15 + w*x^14 + w^2*x^12 + w*x^11 + x^10 + w^2*x^9 + x^8 + x^7 + x^6 + w^2*x^5 + w^2*x^4 + w*x^3 + 1
[2]:  [[35, 27, 3]] quantum code over GF(2^2)
     Subcode of [1]
[3]:  [[36, 27, 3]] quantum code over GF(2^2)
     ExtendCode [2] by 1

    stabilizer matrix:

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

last modified: 2006-04-06

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.