MinT - Best Known (66−31, 66, s)-Nets in Base 2

Best Known (66−31, 66, s)-Nets in Base 2

(66−31, 66, 26)-Net over F₂ — Constructive and digital

Digital (35, 66, 26)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (9, 24, 12)-net over F₂, using
  - net from sequence [i] based on digital (9, 11)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 9 and N(F) ≥ 12, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (11, 42, 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]

(66−31, 66, 29)-Net over F₂ — Digital

Digital (35, 66, 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]

(66−31, 66, 98)-Net in Base 2 — Upper bound on s

There is no (35, 66, 99)-net in base 2, because

1 times m-reduction [i] would yield (35, 65, 99)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁶⁵, 99, S₂, 30), but
  - the linear programming bound shows that MÂ â‰¥ 3339 651673 727309 508069 294080 / 86 822723 > 2⁶⁵ [i]