MinT - Best Known (42, 42+89, s)-Nets in Base 2

Best Known (42, 42+89, s)-Nets in Base 2

(42, 42+89, 33)-Net over F₂ — Constructive and digital

Digital (42, 131, 33)-net over F₂, using

t-expansion [i] based on digital (39, 131, 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]

(42, 42+89, 64)-Net in Base 2 — Upper bound on s

There is no (42, 131, 65)-net in base 2, because

8 times m-reduction [i] would yield (42, 123, 65)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹²³, 65, S₂, 2, 81), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 510 423550 381407 695195 061911 147652 317184 / 41 > 2¹²³ [i]