MinT - Best Known (219−95, 219, s)-Nets in Base 2

Best Known (219−95, 219, s)-Nets in Base 2

(219−95, 219, 63)-Net over F₂ — Constructive and digital

Digital (124, 219, 63)-net over F₂, using

3 times m-reduction [i] based on digital (124, 222, 63)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 70, 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 (54, 152, 42)-net over F₂, using
    - net from sequence [i] based on digital (54, 41)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
        an explicitly constructive algebraic function field [i]

(219−95, 219, 83)-Net over F₂ — Digital

Digital (124, 219, 83)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−95, 219, 391)-Net in Base 2 — Upper bound on s

There is no (124, 219, 392)-net in base 2, because

1 times m-reduction [i] would yield (124, 218, 392)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 421350 316910 179475 173433 016955 375185 994890 554659 163582 684187 480592 > 2²¹⁸ [i]