MinT - Best Known (260−109, 260, s)-Nets in Base 2

Best Known (260−109, 260, s)-Nets in Base 2

(260−109, 260, 75)-Net over F₂ — Constructive and digital

Digital (151, 260, 75)-net over F₂, using

2¹ times duplication [i] based on digital (150, 259, 75)-net over F₂, using
- t-expansion [i] based on digital (148, 259, 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, 165, 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]

(260−109, 260, 104)-Net over F₂ — Digital

Digital (151, 260, 104)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(260−109, 260, 506)-Net in Base 2 — Upper bound on s

There is no (151, 260, 507)-net in base 2, because

1 times m-reduction [i] would yield (151, 259, 507)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 940437 133888 918158 261292 107082 185769 940459 599099 712707 687076 438436 200374 891402 > 2²⁵⁹ [i]