Bounds on the minimum distance of additive quantum codes

Bounds on [[57,27]]2

lower bound:7
upper bound:10

Construction

Construction of a [[57,27,7]] quantum code:
[1]:  [[58, 28, 8]] quantum code over GF(2^2)
     quasicyclic code of length 58 stacked to height 2 with 4 generating polynomials
[2]:  [[57, 29, 7]] quantum code over GF(2^2)
     Shortening of the stabilizer code of [1] at 58
[3]:  [[57, 27, 7]] quantum code over GF(2^2)
     Subcode of [2]

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

last modified: 2006-04-03

Notes


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