MinT - Best Known (219−89, 219, s)-Nets in Base 2

Best Known (219−89, 219, s)-Nets in Base 2

(219−89, 219, 67)-Net over F₂ — Constructive and digital

Digital (130, 219, 67)-net over F₂, using

3 times m-reduction [i] based on digital (130, 222, 67)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 85, 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, 137, 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]

(219−89, 219, 96)-Net over F₂ — Digital

Digital (130, 219, 96)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−89, 219, 472)-Net in Base 2 — Upper bound on s

There is no (130, 219, 473)-net in base 2, because

1 times m-reduction [i] would yield (130, 218, 473)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 421817 206937 097312 688581 996473 253391 544582 288180 145676 794945 392280 > 2²¹⁸ [i]