MinT - Best Known (219−178, 219, s)-Nets in Base 2

Best Known (219−178, 219, s)-Nets in Base 2

(219−178, 219, 33)-Net over F₂ — Constructive and digital

Digital (41, 219, 33)-net over F₂, using

t-expansion [i] based on digital (39, 219, 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]

(219−178, 219, 51)-Net in Base 2 — Upper bound on s

There is no (41, 219, 52)-net in base 2, because

19 times m-reduction [i] would yield (41, 200, 52)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2²⁰⁰, 52, S₂, 4, 159), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 9 641628 265553 941653 251772 554046 975615 133217 962696 757011 808256 / 5 > 2²⁰⁰ [i]