MinT - Best Known (41, 41+108, s)-Nets in Base 2

Best Known (41, 41+108, s)-Nets in Base 2

(41, 41+108, 33)-Net over F₂ — Constructive and digital

Digital (41, 149, 33)-net over F₂, using

t-expansion [i] based on digital (39, 149, 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]

(41, 41+108, 63)-Net in Base 2 — Upper bound on s

There is no (41, 149, 64)-net in base 2, because

29 times m-reduction [i] would yield (41, 120, 64)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹²⁰, 64, S₂, 2, 79), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 7 310753 976817 037300 970938 831541 895168 / 5 > 2¹²⁰ [i]