MinT - Best Known (220−85, 220, s)-Nets in Base 2

Best Known (220−85, 220, s)-Nets in Base 2

(220−85, 220, 75)-Net over F₂ — Constructive and digital

Digital (135, 220, 75)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (39, 81, 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, 139, 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]

(220−85, 220, 108)-Net over F₂ — Digital

Digital (135, 220, 108)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(220−85, 220, 553)-Net in Base 2 — Upper bound on s

There is no (135, 220, 554)-net in base 2, because

1 times m-reduction [i] would yield (135, 219, 554)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 886502 114973 317957 890365 387529 869389 218668 522784 641408 294135 985128 > 2²¹⁹ [i]