MinT - Best Known (234−109, 234, s)-Nets in Base 2

Best Known (234−109, 234, s)-Nets in Base 2

(234−109, 234, 59)-Net over F₂ — Constructive and digital

Digital (125, 234, 59)-net over F₂, using

3 times m-reduction [i] based on digital (125, 237, 59)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 71, 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, 166, 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]

(234−109, 234, 80)-Net over F₂ — Digital

Digital (125, 234, 80)-net over F₂, using

t-expansion [i] based on digital (121, 234, 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]

(234−109, 234, 269)-Net in Base 2 — Upper bound on s

There is no (125, 234, 270)-net in base 2, because

1 times m-reduction [i] would yield (125, 233, 270)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²³³, 270, S₂, 108), but
  - the linear programming bound shows that MÂ â‰¥ 1 805716 963838 191291 276449 773987 529933 551024 538218 073842 142319 599295 484110 749441 196032 / 124 860938 893295 > 2²³³ [i]