Bounds on the minimum distance of additive quantum codes

Bounds on [[81,17]]2

lower bound:13
upper bound:23

Construction

Construction of a [[81,17,13]] quantum code:
[1]:  [[85, 13, 16]] quantum code over GF(2^2)
     cyclic code of length 85 with generating polynomials [
w^2*x^84 + w^2*x^83 + x^82 + w^2*x^81 + w^2*x^80 + w^2*x^79 + w*x^78 + x^77 + w*x^76 + x^75 + w^2*x^74 + w*x^72 + w*x^71 + w*x^70 + w*x^69 + x^68 + w^2*x^67 + w^2*x^66 + x^63 + w*x^62 + w*x^61 + x^60 + w^2*x^58 + x^57 + w^2*x^55 + w*x^54 + x^53 + w*x^52 + x^51 + w^2*x^50 + w*x^49 + w^2*x^48 + w*x^47 + w*x^46 + x^45 + x^43 + x^40 + x^39 + x^38 + w*x^37 + x^36 + 1,
x^84 + x^83 + w*x^82 + x^81 + x^80 + x^79 + w^2*x^78 + w*x^77 + w^2*x^76 + w*x^75 + x^74 + w^2*x^72 + w^2*x^71 + w^2*x^70 + w^2*x^69 + w*x^68 + x^67 + x^66 + w*x^63 + w^2*x^62 + w^2*x^61 + w*x^60 + x^58 + w*x^57 + x^55 + w^2*x^54 + w*x^53 + w^2*x^52 + w*x^51 + x^50 + w^2*x^49 + x^48 + w^2*x^47 + w^2*x^46 + w*x^45 + w*x^43 + w*x^40 + w*x^39 + w*x^38 + w^2*x^37 + w*x^36 + w
]
[2]:  [[81, 17, 13]] quantum code over GF(2^2)
     Shortening of the stabilizer code of [1] at { 1, 2, 7, 18 }

    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 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 1 0 0 0 0 0 1 1 1 0 0 0 1 0 0 1 1 0 1 1 0 1 1 1|0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 1 1 0 1 0 1 1 1 0 0 1 0 1 1 1 1 1 0 1 0 0 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0 1 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 0 0 0 0 0 0 0 0 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 1 1 1 0 0 1 1 1 1 1 1 1 0 0 0 1 0 0 0 0 0 0 1 1 1|0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 1 1 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 0 1 0 1 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 1 1 1 0 0 0 0 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 0 0 0 0 0 0 0 0 0 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 1 0 1 0 0 1 1 1 0 1 0 1 1 1 1 1 0 1 1 1 1 1|0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 1 0 1 0 0 0 1 0 1 1 1 1 0 0 0 1 1 0 1 0 1 0 1 1 0 0 1 0 1 0 0 0 1 1 1 0 1 0 1 0 0 0 1 1 0 1 0 0 1 0 0 1 0 0 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 1 0 1 1 0 0 1 1 1 1 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 1|0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 1 1 1 0 1 0 0 1 1 1 1 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 1 1 1 0 0 0 1 1 0 1 0 1 1 0 1 1 1 0 0 1 0 0 0 0 0 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 0 0 0 0 0 0 0 0 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 1 1 1 1 0 0 0 0 1 1 0 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 1 1 1 0 0 1 1 1 1 0 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 0 0 0 1 1 0 1 1 1 1 0 1 0 1 1 0 0 0 0 1 0 1 0 0 1 1 1 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 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 1 1 1 0 0 0 0 1 1 0 1 1 1 1 1 1 1|0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 1 0 1 0 1 1 0 0 1 0 1 1 1 1 0 1 0 1 0 1 0 1 0 0 0 0 1 0 0 0 1 1 0 1 1 0 1 1 1 0 0 1 0 0 0 1 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 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 1 0 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 0 0 0 1 1|0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 1 1 1 0 0 1 0 1 1 1 1 0 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 0 0 0 0 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 1 1 0 1 1 1 1 0 0 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 1 0 0 1 1 1 0 0 1 0 0 1 0 0 1 1 1 0 0 0 1 0 0 0 1 0 1 1 1 0 0 1 0 1 1 1 1 1 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 0 1 0 0 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 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 1 1 0 0 0 0 1 1 1 1 0 0 0 1 0 0 0 1 1 0 1 0 1 0|0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 1 1 1 1 0 1 1 0 0 1 0 1 1 0 1 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 1 1 1 0 1 1 1 0 0 1 0 1 1 0 1 1 1 1 0 0 1 1 1 1 0 1 1 1 1]
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      [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 0 0 0 0 0 1 1 0 1 1 1 1 1 1 1 1 1 0 1 1 0 1 0 1 1 1 1 1 0 0 0 1 1 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 0 0 0 1 0 1 1 0 1 0 1 0 0 1 0 1 1 1 0 0 0 0 1 1 0 1 0 1 0 0 1 1 0 0 0 1 0 0 0 1 1 0 1 0 0 1 0 0 1 0 0 1 1 1 0 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 0 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 1 1 1 1 1 1 1 1 0 1 1 0 1 0 1 1 1 1 1 0 0 0 1 1|0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 1 0 1 1 1 0 1 1 0 1 1 0 1 0 1 1 1 1 1 0 1 0 1 0 1 0 1 1 0 1 0 0 0 1 1 1 1 1 1 0 0 1 1 1 1 0 1 1 0 0 1 0 0 1 0 1 0 0 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 0 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 1 1 1 1 0 0 0 1 1 0 0 1 0 0 1 1 0 0 1 0 1 1 0 1|0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 1 0 1 1 1 1 1 1 1 1 1 0 1 1 1 0 0 0 1 0 1 0 0 0 1 1 0 1 0 1 0 1 0 0 0 1 1 0 1 1 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 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 0 0 0 0 0 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 1 1 0 1 1 0 1 1 1 0 1 1 1 1 0 1 0 0 1 0 1 0|0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 1 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 1 1 0 1 1 1 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 0 0 0 1]
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      [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 0 1 1 0 1 0 1 0 0 1 0 0 1 1 1 0 0 1 1 1 0 1 0 0 0 0 1 1 1|0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 0 1 1 0 0 0 1 1 0 0 0 1 1 1 1 1]
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      [0 0 0 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 1 1 0 0 1 0 0 0 1 0 0 0 0 0 1 1 1 1 0 0 1 0 0 1 0 0 0 1 0|0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 0 0 1 1 1 1 1 0 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 1 1 0 1 0 1 1 1 0 0 0 0 1 1 0 1 1 1 0 0 0 1 1 1 0 1 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0]
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last modified: 2006-04-03

Notes


This page is maintained by Markus Grassl (codes@codetables.de). Last change: 23.10.2014