MinT - Best Known (65−31, 65, s)-Nets in Base 2

Best Known (65−31, 65, s)-Nets in Base 2

(65−31, 65, 25)-Net over F₂ — Constructive and digital

Digital (34, 65, 25)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (8, 23, 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, 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]

(65−31, 65, 28)-Net over F₂ — Digital

Digital (34, 65, 28)-net over F₂, using

t-expansion [i] based on digital (33, 65, 28)-net over F₂, using
- net from sequence [i] based on digital (33, 27)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 33 and N(F) ≥ 28, using
    - Drinfeld modules of rank 1 [i]

(65−31, 65, 92)-Net in Base 2 — Upper bound on s

There is no (34, 65, 93)-net in base 2, because

1 times m-reduction [i] would yield (34, 64, 93)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2⁶⁴, 93, S₂, 30), but
  - the linear programming bound shows that MÂ â‰¥ 2232 046957 120771 480436 604928 / 99 673145 > 2⁶⁴ [i]