MinT - Best Known (165, 165+49, s)-Nets in Base 2

Best Known (165, 165+49, s)-Nets in Base 2

(165, 165+49, 200)-Net over F₂ — Constructive and digital

Digital (165, 214, 200)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 25, 5)-net over F₂, using
  - net from sequence [i] based on digital (1, 4)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 1 and N(F) ≥ 5, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (140, 189, 195)-net over F₂, using
  - trace code for nets [i] based on digital (14, 63, 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, 165+49, 373)-Net over F₂ — Digital

Digital (165, 214, 373)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(165, 165+49, 4567)-Net in Base 2 — Upper bound on s

There is no (165, 214, 4568)-net in base 2, because

1 times m-reduction [i] would yield (165, 213, 4568)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 13216 847207 609169 628692 977251 993310 458980 361203 804403 338845 475848 > 2²¹³ [i]