MinT - Best Known (165−65, 165, s)-Nets in Base 2

Best Known (165−65, 165, s)-Nets in Base 2

(165−65, 165, 66)-Net over F₂ — Constructive and digital

Digital (100, 165, 66)-net over F₂, using

5 times m-reduction [i] based on digital (100, 170, 66)-net over F₂, using
- trace code for nets [i] based on digital (15, 85, 33)-net over F₄, using
  - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]

(165−65, 165, 82)-Net over F₂ — Digital

Digital (100, 165, 82)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(165−65, 165, 401)-Net in Base 2 — Upper bound on s

There is no (100, 165, 402)-net in base 2, because

1 times m-reduction [i] would yield (100, 164, 402)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 25 099485 825071 780734 540255 790748 377170 748459 492508 > 2¹⁶⁴ [i]