MinT - Best Known (232−106, 232, s)-Nets in Base 2

Best Known (232−106, 232, s)-Nets in Base 2

(232−106, 232, 62)-Net over F₂ — Constructive and digital

Digital (126, 232, 62)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (19, 72, 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, 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]

(232−106, 232, 81)-Net over F₂ — Digital

Digital (126, 232, 81)-net over F₂, using

net from sequence [i] based on digital (126, 80)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 126 and N(F) ≥ 81, using
  - Drinfeld modules of rank 1 [i]

(232−106, 232, 278)-Net in Base 2 — Upper bound on s

There is no (126, 232, 279)-net in base 2, because

extracting embedded orthogonal array [i] would yield OA(2²³², 279, S₂, 106), but
- the linear programming bound shows that MÂ â‰¥ 34 039058 206152 067616 658014 815118 671716 256151 607742 433060 377115 839120 093330 029911 474176 / 3903 604614 609375 > 2²³² [i]