MinT - Best Known (227−102, 227, s)-Nets in Base 2

Best Known (227−102, 227, s)-Nets in Base 2

(227−102, 227, 62)-Net over F₂ — Constructive and digital

Digital (125, 227, 62)-net over F₂, using

2 times m-reduction [i] based on digital (125, 229, 62)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (19, 71, 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, 158, 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]

(227−102, 227, 80)-Net over F₂ — Digital

Digital (125, 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]

(227−102, 227, 283)-Net in Base 2 — Upper bound on s

There is no (125, 227, 284)-net in base 2, because

extracting embedded orthogonal array [i] would yield OA(2²²⁷, 284, S₂, 102), but
- adding a parity check bit [i] would yield OA(2²²⁸, 285, S₂, 103), but
  - the linear programming bound shows that MÂ â‰¥ 4 034161 780580 224571 807834 810043 778367 016643 680872 935104 289259 268042 795950 972450 751656 230912 / 8986 311777 879741 102525 > 2²²⁸ [i]