MinT - Best Known (249−99, 249, s)-Nets in Base 2

Best Known (249−99, 249, s)-Nets in Base 2

(249−99, 249, 76)-Net over F₂ — Constructive and digital

Digital (150, 249, 76)-net over F₂, using

3 times m-reduction [i] based on digital (150, 252, 76)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (45, 96, 34)-net over F₂, using
    - net from sequence [i] based on digital (45, 33)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 1 place with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]
  2. digital (54, 156, 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]

(249−99, 249, 113)-Net over F₂ — Digital

Digital (150, 249, 113)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(249−99, 249, 568)-Net in Base 2 — Upper bound on s

There is no (150, 249, 569)-net in base 2, because

1 times m-reduction [i] would yield (150, 248, 569)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 462 936530 048487 535394 752219 495318 188609 647655 973349 501960 843303 935910 414286 > 2²⁴⁸ [i]