MinT - Best Known (69−33, 69, s)-Nets in Base 2

Best Known (69−33, 69, s)-Nets in Base 2

(69−33, 69, 26)-Net over F₂ — Constructive and digital

Digital (36, 69, 26)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (9, 25, 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, 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]

(69−33, 69, 30)-Net over F₂ — Digital

Digital (36, 69, 30)-net over F₂, using

net from sequence [i] based on digital (36, 29)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 36 and N(F) ≥ 30, using
  - Drinfeld modules of rank 1 [i]

(69−33, 69, 95)-Net in Base 2 — Upper bound on s

There is no (36, 69, 96)-net in base 2, because

1 times m-reduction [i] would yield (36, 68, 96)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁶⁸, 96, S₂, 32), but
  - the linear programming bound shows that MÂ â‰¥ 1 599539 177775 615668 865588 002816 / 5051 366145 > 2⁶⁸ [i]