MinT - Best Known (225−101, 225, s)-Nets in Base 2

Best Known (225−101, 225, s)-Nets in Base 2

(225−101, 225, 62)-Net over F₂ — Constructive and digital

Digital (124, 225, 62)-net over F₂, using

1 times m-reduction [i] based on digital (124, 226, 62)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (19, 70, 20)-net over F₂, using
    - net from sequence [i] based on digital (19, 19)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 19 and N(F) ≥ 20, using
        an explicitly constructive algebraic function field [i]
  2. digital (54, 156, 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]

(225−101, 225, 80)-Net over F₂ — Digital

Digital (124, 225, 80)-net over F₂, using

t-expansion [i] based on digital (121, 225, 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]

(225−101, 225, 285)-Net in Base 2 — Upper bound on s

There is no (124, 225, 286)-net in base 2, because

1 times m-reduction [i] would yield (124, 224, 286)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²²⁴, 286, S₂, 100), but
  - the linear programming bound shows that MÂ â‰¥ 170771 802709 288542 952317 268135 297071 192680 292267 039441 199237 367446 827296 075719 027611 140096 / 6019 803089 429815 761075 > 2²²⁴ [i]