MinT - Best Known (241−99, 241, s)-Nets in Base 2

Best Known (241−99, 241, s)-Nets in Base 2

(241−99, 241, 75)-Net over F₂ — Constructive and digital

Digital (142, 241, 75)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (39, 88, 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, 153, 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]

(241−99, 241, 102)-Net over F₂ — Digital

Digital (142, 241, 102)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(241−99, 241, 500)-Net in Base 2 — Upper bound on s

There is no (142, 241, 501)-net in base 2, because

1 times m-reduction [i] would yield (142, 240, 501)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1 793682 816306 824433 882695 106264 090491 463165 724678 160376 619904 145633 440082 > 2²⁴⁰ [i]