MinT - Best Known (35, 68, s)-Nets in Base 2

Best Known (35, 68, s)-Nets in Base 2

(35, 68, 25)-Net over F₂ — Constructive and digital

Digital (35, 68, 25)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (8, 24, 11)-net over F₂, using
  - net from sequence [i] based on digital (8, 10)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 8 and N(F) ≥ 11, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (11, 44, 14)-net over F₂, using
  - net from sequence [i] based on digital (11, 13)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 11 and N(F) ≥ 14, using
      - an explicitly constructive algebraic function field [i]

(35, 68, 29)-Net over F₂ — Digital

Digital (35, 68, 29)-net over F₂, using

net from sequence [i] based on digital (35, 28)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 35 and N(F) ≥ 29, using
  - a curve from the manYPoints database [i]

(35, 68, 90)-Net in Base 2 — Upper bound on s

There is no (35, 68, 91)-net in base 2, because

1 times m-reduction [i] would yield (35, 67, 91)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁶⁷, 91, S₂, 32), but
  - the linear programming bound shows that MÂ â‰¥ 16 438557 726869 234988 875776 / 92225 > 2⁶⁷ [i]