MinT - Best Known (231−188, 231, s)-Nets in Base 2

Best Known (231−188, 231, s)-Nets in Base 2

(231−188, 231, 33)-Net over F₂ — Constructive and digital

Digital (43, 231, 33)-net over F₂, using

t-expansion [i] based on digital (39, 231, 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]

(231−188, 231, 34)-Net over F₂ — Digital

Digital (43, 231, 34)-net over F₂, using

net from sequence [i] based on digital (43, 33)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 43 and N(F) ≥ 34, using
  - a curve from the manYPoints database [i]

(231−188, 231, 53)-Net in Base 2 — Upper bound on s

There is no (43, 231, 54)-net in base 2, because

22 times m-reduction [i] would yield (43, 209, 54)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2²⁰⁹, 54, S₂, 4, 166), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 171132 473961 405428 384116 794985 964452 518204 530025 892305 788255 338496 / 167 > 2²⁰⁹ [i]