MinT - Best Known (227−105, 227, s)-Nets in Base 2

Best Known (227−105, 227, s)-Nets in Base 2

(227−105, 227, 59)-Net over F₂ — Constructive and digital

Digital (122, 227, 59)-net over F₂, using

1 times m-reduction [i] based on digital (122, 228, 59)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 68, 17)-net over F₂, using
    - net from sequence [i] based on digital (15, 16)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 15 and N(F) ≥ 17, using
        an explicitly constructive algebraic function field [i]
  2. digital (54, 160, 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]

(227−105, 227, 80)-Net over F₂ — Digital

Digital (122, 227, 80)-net over F₂, using

t-expansion [i] based on digital (121, 227, 80)-net over F₂, using
- net from sequence [i] based on digital (121, 79)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 121 and N(F) ≥ 80, using
    - Drinfeld modules of rank 1 [i]

(227−105, 227, 267)-Net in Base 2 — Upper bound on s

There is no (122, 227, 268)-net in base 2, because

1 times m-reduction [i] would yield (122, 226, 268)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²²⁶, 268, S₂, 104), but
  - the linear programming bound shows that MÂ â‰¥ 148 703935 381500 786165 871722 483184 672184 725384 011978 704151 839426 381785 270960 662161 391616 / 1 078465 608545 441415 > 2²²⁶ [i]