Bounds on the minimum distance of additive quantum codes

Bounds on [[57,35]]₂

lower bound:	5
upper bound:	8

Construction

Construction of a [[57,35,5]] quantum code:
[1]:  [[87, 59, 6]] quantum code over GF(2^2)
     QuasiCyclicCode of length 87 with generating polynomials: x^28 + w*x^27 + x^26 + w*x^24 + w^2*x^22 + w*x^20 + w*x^19 + w*x^18 + w*x^17 + w*x^16 + w^2*x^15 + x^14 + w*x^12 + w*x^11 + x^10 + x^9 + w*x^7 + w*x^6 + w*x^5 + w^2*x^4 + w*x^3 + w*x^2 + x + w^2,  x^27 + w^2*x^24 + w*x^23 + x^22 + x^20 + w^2*x^17 + x^16 + w^2*x^14 + w^2*x^13 + w*x^12 + w^2*x^11 + w^2*x^10 + x^9 + w*x^8 + w*x^6 + w*x^5 + x^4 + w^2*x^3 + w*x + w^2,  w^2*x^28 + x^27 + w^2*x^25 + x^24 + w^2*x^23 + w*x^21 + w^2*x^20 + w^2*x^19 + x^18 + w*x^16 + x^14 + w^2*x^13 + w*x^12 + w*x^10 + x^9 + w^2*x^8 + w^2*x^7 + w*x^6 + x^5 + x^4 + x^3 + x^2 + w*x + 1
[2]:  [[84, 62, 5]] quantum code over GF(2^2)
     Shortening of the stabilizer code of [1] at { 3, 24, 30 }
[3]:  [[57, 35, 5]] quantum code over GF(2^2)
     Shortening of [2] at { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27 }

    stabilizer matrix:

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

last modified: 2006-04-03

Notes

All codes establishing the lower bounds where constructed using MAGMA.
Most upper bounds on qubit codes for n≤100 are based on a MAGMA program by Eric Rains.
For n>100, the upper bounds on qubit codes are weak (and not necessarily monotone in k).
Some additional information can be found in the book by Nebe, Rains, and Sloane.
My apologies to all authors that have contributed codes to this table for not giving specific credits.

This page is maintained by Markus Grassl (codes@codetables.de). Last change: 10.06.2024

Bounds on the minimum distance of additive quantum codes

Bounds on [[57,35]]2

Construction

Notes

Bounds on [[57,35]]₂