MinT - Best Known (72−34, 72, s)-Nets in Base 2

Best Known (72−34, 72, s)-Nets in Base 2

(72−34, 72, 27)-Net over F₂ — Constructive and digital

Digital (38, 72, 27)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (6, 23, 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, 49, 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]

(72−34, 72, 30)-Net over F₂ — Digital

Digital (38, 72, 30)-net over F₂, using

t-expansion [i] based on digital (36, 72, 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]

(72−34, 72, 96)-Net in Base 2 — Upper bound on s

There is no (38, 72, 97)-net in base 2, because

extracting embedded orthogonal array [i] would yield OA(2⁷², 97, S₂, 34), but
- the linear programming bound shows that MÂ â‰¥ 84 473691 645572 213582 594048 / 17545 > 2⁷² [i]