MinT - Best Known (241−105, 241, s)-Nets in Base 2

Best Known (241−105, 241, s)-Nets in Base 2

(241−105, 241, 67)-Net over F₂ — Constructive and digital

Digital (136, 241, 67)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (39, 91, 33)-net over F₂, using
  - net from sequence [i] based on digital (39, 32)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 39 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]
2. digital (45, 150, 34)-net over F₂, using
  - net from sequence [i] based on digital (45, 33)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 1 place with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]

(241−105, 241, 89)-Net over F₂ — Digital

Digital (136, 241, 89)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(241−105, 241, 423)-Net in Base 2 — Upper bound on s

There is no (136, 241, 424)-net in base 2, because

1 times m-reduction [i] would yield (136, 240, 424)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1 889762 752135 042699 301329 522563 648619 619374 163324 895050 097307 578464 333512 > 2²⁴⁰ [i]