MinT - Best Known (259−109, 259, s)-Nets in Base 2

Best Known (259−109, 259, s)-Nets in Base 2

(259−109, 259, 75)-Net over F₂ — Constructive and digital

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]

(259−109, 259, 103)-Net over F₂ — Digital

Digital (150, 259, 103)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(259−109, 259, 499)-Net in Base 2 — Upper bound on s

There is no (150, 259, 500)-net in base 2, because

1 times m-reduction [i] would yield (150, 258, 500)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 487230 787584 262402 207093 660974 684999 816528 711078 807334 456439 367476 455110 021376 > 2²⁵⁸ [i]