MinT - Best Known (256−107, 256, s)-Nets in Base 2

Best Known (256−107, 256, s)-Nets in Base 2

(256−107, 256, 75)-Net over F₂ — Constructive and digital

Digital (149, 256, 75)-net over F₂, using

t-expansion [i] based on digital (148, 256, 75)-net over F₂, using
- 2 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]

(256−107, 256, 103)-Net over F₂ — Digital

Digital (149, 256, 103)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(256−107, 256, 503)-Net in Base 2 — Upper bound on s

There is no (149, 256, 504)-net in base 2, because

1 times m-reduction [i] would yield (149, 255, 504)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 58513 484147 502012 415981 489540 035122 511505 595537 554386 952006 015149 460122 987397 > 2²⁵⁵ [i]