Bounds on the minimum distance of additive quantum codes

Bounds on [[41,7]]2

lower bound:9
upper bound:13

Construction

Construction of a [[41,7,9]] quantum code:
[1]:  [[40, 6, 10]] quantum code over GF(2^2)
     QuasiCyclicCode of length 40 with generating polynomials: w^2*x^18 + w^2*x^17 + w^2*x^16 + w*x^15 + x^3,  w^2*x^19 + w*x^18 + x^17 + w*x^15 + w*x^14 + w*x^12 + x^11 + w*x^10 + x^9 + w*x^7 + w*x^5
[2]:  [[39, 7, 9]] quantum code over GF(2^2)
     Shortening of the stabilizer code of [1] at { 40 }
[3]:  [[41, 7, 9]] quantum code over GF(2^2)
     ExtendCode [2] by 2

    stabilizer matrix:

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

last modified: 2023-01-04

Notes


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