MinT - Best Known (165−39, 165, s)-Nets in Base 2

Best Known (165−39, 165, s)-Nets in Base 2

(165−39, 165, 195)-Net over F₂ — Constructive and digital

Digital (126, 165, 195)-net over F₂, using

3 times m-reduction [i] based on digital (126, 168, 195)-net over F₂, using
- trace code for nets [i] based on digital (14, 56, 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]

(165−39, 165, 276)-Net over F₂ — Digital

Digital (126, 165, 276)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(165−39, 165, 3116)-Net in Base 2 — Upper bound on s

There is no (126, 165, 3117)-net in base 2, because

1 times m-reduction [i] would yield (126, 164, 3117)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 23 409999 817020 901304 621039 418795 900161 734644 045824 > 2¹⁶⁴ [i]