MinT - Best Known (215−93, 215, s)-Nets in Base 2

Best Known (215−93, 215, s)-Nets in Base 2

(215−93, 215, 63)-Net over F₂ — Constructive and digital

Digital (122, 215, 63)-net over F₂, using

1 times m-reduction [i] based on digital (122, 216, 63)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 68, 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, 148, 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]

(215−93, 215, 82)-Net over F₂ — Digital

Digital (122, 215, 82)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(215−93, 215, 388)-Net in Base 2 — Upper bound on s

There is no (122, 215, 389)-net in base 2, because

1 times m-reduction [i] would yield (122, 214, 389)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 27883 492075 801845 344331 394924 827207 329729 971414 458775 889138 344320 > 2²¹⁴ [i]