MinT - Best Known (218−81, 218, s)-Nets in Base 2

Best Known (218−81, 218, s)-Nets in Base 2

(218−81, 218, 75)-Net over F₂ — Constructive and digital

Digital (137, 218, 75)-net over F₂, using

7 times m-reduction [i] based on digital (137, 225, 75)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 83, 33)-net over F₂, using
    - net from sequence [i] based on digital (39, 32)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 39 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
  2. digital (54, 142, 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]

(218−81, 218, 84)-Net in Base 2 — Constructive

(137, 218, 84)-net in base 2, using

2 times m-reduction [i] based on (137, 220, 84)-net in base 2, using
- trace code for nets [i] based on (27, 110, 42)-net in base 4, using
  - net from sequence [i] based on (27, 41)-sequence in base 4, using
    - base expansion [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]

(218−81, 218, 117)-Net over F₂ — Digital

Digital (137, 218, 117)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(218−81, 218, 620)-Net in Base 2 — Upper bound on s

There is no (137, 218, 621)-net in base 2, because

1 times m-reduction [i] would yield (137, 217, 621)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 220383 599399 927051 498385 383648 654046 893277 910187 583965 213017 801041 > 2²¹⁷ [i]