MinT - Best Known (251−103, 251, s)-Nets in Base 2

Best Known (251−103, 251, s)-Nets in Base 2

(251−103, 251, 75)-Net over F₂ — Constructive and digital

Digital (148, 251, 75)-net over F₂, using

7 times m-reduction [i] based on digital (148, 258, 75)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 94, 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, 164, 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]

(251−103, 251, 106)-Net over F₂ — Digital

Digital (148, 251, 106)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(251−103, 251, 521)-Net in Base 2 — Upper bound on s

There is no (148, 251, 522)-net in base 2, because

1 times m-reduction [i] would yield (148, 250, 522)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1846 682192 286208 581851 055891 202113 060618 563861 704066 150795 932794 436238 647552 > 2²⁵⁰ [i]