MinT - Best Known (133−89, 133, s)-Nets in Base 2

Best Known (133−89, 133, s)-Nets in Base 2

(133−89, 133, 33)-Net over F₂ — Constructive and digital

Digital (44, 133, 33)-net over F₂, using

t-expansion [i] based on digital (39, 133, 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]

(133−89, 133, 34)-Net over F₂ — Digital

Digital (44, 133, 34)-net over F₂, using

t-expansion [i] based on digital (43, 133, 34)-net over F₂, using
- net from sequence [i] based on digital (43, 33)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 43 and N(F) ≥ 34, using
    - a curve from the manYPoints database [i]

(133−89, 133, 67)-Net in Base 2 — Upper bound on s

There is no (44, 133, 68)-net in base 2, because

5 times m-reduction [i] would yield (44, 128, 68)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹²⁸, 68, S₂, 2, 84), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 29944 848289 042584 784776 965453 995602 608128 / 85 > 2¹²⁸ [i]