MinT - Best Known (40, 40+172, s)-Nets in Base 2

Best Known (40, 40+172, s)-Nets in Base 2

(40, 40+172, 33)-Net over F₂ — Constructive and digital

Digital (40, 212, 33)-net over F₂, using

t-expansion [i] based on digital (39, 212, 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]

(40, 40+172, 50)-Net in Base 2 — Upper bound on s

There is no (40, 212, 51)-net in base 2, because

17 times m-reduction [i] would yield (40, 195, 51)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹⁹⁵, 51, S₂, 4, 155), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 2 021226 758794 511205 955124 194272 868585 984958 453117 419113 152512 / 39 > 2¹⁹⁵ [i]