Bounds on the minimum distance of additive quantum codes

Bounds on [[48,33]]2

lower bound:4
upper bound:5

Construction

Construction of a [[48,33,4]] quantum code:
[1]:  [[65, 51, 4]] quantum code over GF(2^2)
     QuasiCyclicCode of length 65 stacked to height 2 with generating polynomials: w^2*x^12 + x^11 + w*x^10 + x^8 + w^2*x^7 + w^2*x^6 + x^5 + x^4 + w*x,  x^11 + w^2*x^10 + w^2*x^8 + w*x^7 + x^6 + w*x^5 + w*x^4 + w*x^3 + w^2*x^2 + w^2*x + w,  x^10 + w^2*x^9 + w^2*x^8 + x^7 + w*x^6 + w^2*x^5 + w^2*x^4 + x^2 + w*x + 1,  w^2*x^12 + w*x^11 + w*x^10 + w^2*x^9 + x^7 + x^6 + x^5 + w^2*x + 1,  w*x^11 + w*x^10 + w^2*x^8 + w*x^7 + w*x^6 + x^5 + w^2*x^3 + x + 1,  x^12 + w*x^11 + w^2*x^10 + w*x^8 + x^7 + x^6 + w*x^5 + w*x^4 + w^2*x,  w*x^11 + x^10 + x^8 + w^2*x^7 + w*x^6 + w^2*x^5 + w^2*x^4 + w^2*x^3 + x^2 + x + w^2,  w*x^10 + x^9 + x^8 + w*x^7 + w^2*x^6 + x^5 + x^4 + w*x^2 + w^2*x + w,  x^12 + w^2*x^11 + w^2*x^10 + x^9 + w*x^7 + w*x^6 + w*x^5 + x + w,  w^2*x^11 + w^2*x^10 + x^8 + w^2*x^7 + w^2*x^6 + w*x^5 + x^3 + w*x + w
[2]:  [[48, 34, 4]] quantum code over GF(2^2)
     Shortening of [1] at { 5, 6, 9, 10, 11, 12, 15, 17, 25, 31, 32, 33, 36, 39, 44, 46, 54 }
[3]:  [[48, 33, 4]] quantum code over GF(2^2)
     Subcode of [2]

    stabilizer matrix:

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

last modified: 2006-04-03

Notes


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