MinT - Best Known (138, 138+101, s)-Nets in Base 2

Best Known (138, 138+101, s)-Nets in Base 2

(138, 138+101, 68)-Net over F₂ — Constructive and digital

Digital (138, 239, 68)-net over F₂, using

1 times m-reduction [i] based on digital (138, 240, 68)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 90, 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 (48, 150, 35)-net over F₂, using
    - net from sequence [i] based on digital (48, 34)-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 2 places 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]

(138, 138+101, 95)-Net over F₂ — Digital

Digital (138, 239, 95)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(138, 138+101, 457)-Net in Base 2 — Upper bound on s

There is no (138, 239, 458)-net in base 2, because

1 times m-reduction [i] would yield (138, 238, 458)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 446628 436874 648264 614538 261376 240416 684712 878670 606630 413676 166853 887344 > 2²³⁸ [i]