MinT - Best Known (233−105, 233, s)-Nets in Base 2

Best Known (233−105, 233, s)-Nets in Base 2

(233−105, 233, 63)-Net over F₂ — Constructive and digital

Digital (128, 233, 63)-net over F₂, using

1 times m-reduction [i] based on digital (128, 234, 63)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 74, 21)-net over F₂, using
    - net from sequence [i] based on digital (21, 20)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 21 and N(F) ≥ 21, 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]

(233−105, 233, 81)-Net over F₂ — Digital

Digital (128, 233, 81)-net over F₂, using

t-expansion [i] based on digital (126, 233, 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]

(233−105, 233, 289)-Net in Base 2 — Upper bound on s

There is no (128, 233, 290)-net in base 2, because

1 times m-reduction [i] would yield (128, 232, 290)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²³², 290, S₂, 104), but
  - the linear programming bound shows that MÂ â‰¥ 31248 188163 173073 124141 552213 221428 911836 972514 149010 859356 835121 130605 760735 093805 023232 / 4 506159 960385 549875 > 2²³² [i]