MinT - Best Known (245−99, 245, s)-Nets in Base 2

Best Known (245−99, 245, s)-Nets in Base 2

(245−99, 245, 75)-Net over F₂ — Constructive and digital

Digital (146, 245, 75)-net over F₂, using

7 times m-reduction [i] based on digital (146, 252, 75)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 92, 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, 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]

(245−99, 245, 107)-Net over F₂ — Digital

Digital (146, 245, 107)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(245−99, 245, 533)-Net in Base 2 — Upper bound on s

There is no (146, 245, 534)-net in base 2, because

1 times m-reduction [i] would yield (146, 244, 534)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 28 754328 507000 137692 408738 504839 441005 476362 775490 983791 468126 208552 078128 > 2²⁴⁴ [i]