Bounds on the minimum distance of additive quantum codes

Bounds on [[56,26]]2

lower bound:7
upper bound:10

Construction

Construction of a [[56,26,7]] quantum code:
[1]:  [[127, 99, 6]] Quantum code over GF(2^2)
     cyclic code of length 127 with generating polynomials [
x^126 + w^2*x^125 + w^2*x^123 + w*x^122 + w*x^121 + x^120 + w^2*x^117 + w*x^116 + w^2*x^115 + w^2*x^113 + x^112 + w^2*x^110 + w*x^109 + x^108 + x^107 + w^2*x^105 + w^2*x^104 + x^103 + w^2*x^101 + w^2*x^100 + x^99 + w*x^98 + w*x^97 + w*x^96 + w^2*x^95 + x^94 + x^93 + w*x^92 + x^91 + w*x^90 + w^2*x^89 + x^87 + x^86 + w*x^85 + w*x^84 + x^83 + x^81 + x^80 + x^79 + x^78 + x^77 + x^76 + w*x^75 + w*x^74 + x^73 + w*x^72 + x^70 + w^2*x^69 + x^68 + x^66 + x^65 + w*x^64 + w^2*x^63 + w*x^62 + w*x^61 + x^60 + w^2*x^58 + w^2*x^57 + x^56 + w^2*x^54 + w*x^53 + x^51 + w^2*x^50 + w^2*x^48 + w^2*x^47 + w*x^45 + w^2*x^44 + x^43 + w*x^42 + w*x^41 + x^40 + x^39 + w^2*x^38 + w^2*x^37 + w^2*x^36 + w*x^35 + x^33 + x^32 + x^31 + w*x^30 + w^2*x^28 + w*x^27 + x^26 + w^2*x^25 + w*x^23 + w*x^22 + w*x^20 + w*x^19 + x^18 + w^2*x^16 + w^2*x^15 + 1,
w^2*x^126 + w^2*x^125 + w^2*x^122 + x^121 + w^2*x^120 + w*x^119 + w*x^117 + w^2*x^116 + w^2*x^115 + x^113 + w*x^112 + w^2*x^111 + w*x^110 + x^108 + w*x^107 + x^105 + w^2*x^104 + w^2*x^102 + w*x^101 + w*x^100 + w^2*x^95 + x^94 + w*x^93 + w^2*x^91 + x^89 + w^2*x^88 + w^2*x^87 + w*x^86 + x^85 + w*x^84 + w^2*x^82 + w*x^81 + x^79 + w*x^78 + w^2*x^77 + w^2*x^76 + w*x^75 + w^2*x^74 + w^2*x^69 + x^68 + x^67 + x^65 + w*x^61 + w*x^60 + w*x^59 + x^58 + w^2*x^57 + w^2*x^55 + w*x^54 + w^2*x^52 + w^2*x^51 + w*x^50 + w^2*x^49 + x^48 + x^47 + w^2*x^46 + x^45 + x^44 + w*x^43 + w*x^42 + x^41 + x^40 + x^39 + x^36 + x^35 + w*x^34 + w^2*x^33 + x^32 + x^31 + w^2*x^30 + w^2*x^28 + x^27 + x^26 + x^25 + w*x^23 + w*x^22 + x^21 + w*x^18 + w*x^17 + x^16 + w^2*x^15 + w*x^14 + w*x^13 + w
]
[2]:  [[128, 98, 7]] Quantum code over GF(2^2)
     standard lengthening of  [1]
[3]:  [[56, 26, 7]] Quantum code over GF(2^2)
     Shortening of [2] at { 4, 6, 7, 8, 11, 14, 16, 20, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 35, 37, 40, 41, 42, 43, 44, 49, 50, 51, 54, 55, 57, 60, 62, 63, 65, 68, 69, 70, 71, 72, 73, 76, 78, 79, 80, 81, 82, 83, 87, 88, 90, 91, 93, 94, 97, 98, 99, 103, 105, 106, 109, 111, 112, 113, 114, 116, 118, 122, 123, 124, 125, 127 }

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

last modified: 2006-04-03

Notes


This page is maintained by Markus Grassl (grassl@ira.uka.de). Last change: 23.10.2014