MinT - Best Known (219−51, 219, s)-Nets in Base 2

Best Known (219−51, 219, s)-Nets in Base 2

(219−51, 219, 195)-Net over F₂ — Constructive and digital

Digital (168, 219, 195)-net over F₂, using

12 times m-reduction [i] based on digital (168, 231, 195)-net over F₂, using
- trace code for nets [i] based on digital (14, 77, 65)-net over F₈, using
  - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
      - the Suzuki function field over F₈ [i]

(219−51, 219, 364)-Net over F₂ — Digital

Digital (168, 219, 364)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−51, 219, 4254)-Net in Base 2 — Upper bound on s

There is no (168, 219, 4255)-net in base 2, because

1 times m-reduction [i] would yield (168, 218, 4255)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 422162 889756 228937 505226 750427 317914 846183 060904 940312 732741 896952 > 2²¹⁸ [i]