MinT - Best Known (237−95, 237, s)-Nets in Base 2

Best Known (237−95, 237, s)-Nets in Base 2

(237−95, 237, 75)-Net over F₂ — Constructive and digital

Digital (142, 237, 75)-net over F₂, using

3 times m-reduction [i] based on digital (142, 240, 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, 152, 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]

(237−95, 237, 106)-Net over F₂ — Digital

Digital (142, 237, 106)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(237−95, 237, 530)-Net in Base 2 — Upper bound on s

There is no (142, 237, 531)-net in base 2, because

1 times m-reduction [i] would yield (142, 236, 531)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 118639 209090 780434 646174 017591 282981 161520 959170 601318 962769 365301 021056 > 2²³⁶ [i]