MinT - Best Known (69−32, 69, s)-Nets in Base 2

Best Known (69−32, 69, s)-Nets in Base 2

(69−32, 69, 27)-Net over F₂ — Constructive and digital

Digital (37, 69, 27)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (6, 22, 10)-net over F₂, using
  - net from sequence [i] based on digital (6, 9)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 6 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (15, 47, 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]

(69−32, 69, 30)-Net over F₂ — Digital

Digital (37, 69, 30)-net over F₂, using

t-expansion [i] based on 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]

(69−32, 69, 101)-Net in Base 2 — Upper bound on s

There is no (37, 69, 102)-net in base 2, because

extracting embedded orthogonal array [i] would yield OA(2⁶⁹, 102, S₂, 32), but
- the linear programming bound shows that MÂ â‰¥ 157615 233778 997728 417366 933504 / 233 882883 > 2⁶⁹ [i]