MinT - Best Known (218−84, 218, s)-Nets in Base 2

Best Known (218−84, 218, s)-Nets in Base 2

(218−84, 218, 70)-Net over F₂ — Constructive and digital

Digital (134, 218, 70)-net over F₂, using

2 times m-reduction [i] based on digital (134, 220, 70)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 64, 21)-net over F₂, using
    - net from sequence [i] based on digital (21, 20)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 21 and N(F) ≥ 21, using
        an explicitly constructive algebraic function field [i]
  2. digital (70, 156, 49)-net over F₂, using
    - net from sequence [i] based on digital (70, 48)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 69, N(F)Â =Â 48, and 1 place with degree 2 [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]

(218−84, 218, 108)-Net over F₂ — Digital

Digital (134, 218, 108)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(218−84, 218, 543)-Net in Base 2 — Upper bound on s

There is no (134, 218, 544)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 443324 460097 982498 399646 830194 672487 951170 330239 957162 397463 572816 > 2²¹⁸ [i]