Bounds on the minimum distance of additive quantum codes

Bounds on [[45,17]]2

lower bound:7
upper bound:10

Construction

Construction of a [[45,17,7]] quantum code:
[1]:  [[85, 57, 7]] Quantum code over GF(2^2)
     cyclic code of length 85 with generating polynomial w*x^84 + w^2*x^83 + w^2*x^82 + w^2*x^81 + w^2*x^80 + w^2*x^79 + x^78 + w^2*x^77 + x^76 + w*x^75 + w^2*x^74 + w*x^73 + w*x^72 + w^2*x^71 + w^2*x^70 + x^68 + x^67 + w*x^66 + x^65 + w*x^64 + w^2*x^63 + w*x^59 + w*x^58 + x^57 + w^2*x^56 + w^2*x^55 + x^54 + w*x^53 + x^52 + x^51 + w^2*x^50 + w*x^49 + w^2*x^48 + w^2*x^47 + x^46 + w*x^45 + w*x^44 + x^43 + w^2*x^42 + w*x^41 + w^2*x^40 + w*x^39 + w*x^38 + w*x^37 + x^36 + w*x^35 + x^33 + w*x^32 + w^2*x^31 + w*x^29 + w^2*x^28 + w*x^27 + w^2*x^26 + w^2*x^25 + x^24 + w*x^22 + x^21 + w^2*x^20 + w*x^19 + w*x^17 + x^16 + w*x^15 + x^14 + 1
[2]:  [[45, 17, 7]] Quantum code over GF(2^2)
     Shortening of [1] at { 1, 2, 7, 9, 10, 13, 14, 17, 23, 24, 26, 27, 28, 30, 31, 32, 39, 42, 44, 45, 47, 55, 56, 57, 60, 63, 64, 65, 66, 67, 68, 69, 70, 72, 76, 77, 78, 80, 82, 84 }

    stabilizer matrix:

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

last modified: 2006-04-03

Notes


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