MinT - Best Known (252−194, 252, s)-Nets in Base 2

Best Known (252−194, 252, s)-Nets in Base 2

(252−194, 252, 42)-Net over F₂ — Constructive and digital

Digital (58, 252, 42)-net over F₂, using

t-expansion [i] based on digital (54, 252, 42)-net over F₂, using
- net from sequence [i] based on digital (54, 41)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
    - an explicitly constructive algebraic function field [i]

(252−194, 252, 74)-Net in Base 2 — Upper bound on s

There is no (58, 252, 75)-net in base 2, because

34 times m-reduction [i] would yield (58, 218, 75)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2²¹⁸, 75, S₂, 3, 160), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 87 619826 668239 579332 667799 032813 799689 320719 373256 860563 586733 309952 / 161 > 2²¹⁸ [i]