MinT - Best Known (18, 18+72, s)-Nets in Base 2

Best Known (18, 18+72, s)-Nets in Base 2

(18, 18+72, 17)-Net over F₂ — Constructive and digital

Digital (18, 90, 17)-net over F₂, using

t-expansion [i] based on digital (15, 90, 17)-net over F₂, using
- net from sequence [i] based on digital (15, 16)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 15 and N(F) ≥ 17, using
    - an explicitly constructive algebraic function field [i]

(18, 18+72, 18)-Net over F₂ — Digital

Digital (18, 90, 18)-net over F₂, using

net from sequence [i] based on digital (18, 17)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 18 and N(F) ≥ 18, using
  - a curve from the manYPoints database [i]

(18, 18+72, 27)-Net in Base 2 — Upper bound on s

There is no (18, 90, 28)-net in base 2, because

13 times m-reduction [i] would yield (18, 77, 28)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2⁷⁷, 28, S₂, 3, 59), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 2 417851 639229 258349 412352 / 15 > 2⁷⁷ [i]