MinT - Best Known (40, 40+209, s)-Nets in Base 2

Best Known (40, 40+209, s)-Nets in Base 2

(40, 40+209, 33)-Net over F₂ — Constructive and digital

Digital (40, 249, 33)-net over F₂, using

t-expansion [i] based on digital (39, 249, 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+209, 49)-Net in Base 2 — Upper bound on s

There is no (40, 249, 50)-net in base 2, because

9 times m-reduction [i] would yield (40, 240, 50)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2²⁴⁰, 50, S₂, 5, 200), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 669 635037 551007 660912 069752 781566 117498 616396 915859 585128 088750 289902 895104 / 335 > 2²⁴⁰ [i]