MinT - Best Known (236−109, 236, s)-Nets in Base 2

Best Known (236−109, 236, s)-Nets in Base 2

(236−109, 236, 62)-Net over F₂ — Constructive and digital

Digital (127, 236, 62)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (19, 73, 20)-net over F₂, using
  - net from sequence [i] based on digital (19, 19)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 19 and N(F) ≥ 20, using
      - an explicitly constructive algebraic function field [i]
2. digital (54, 163, 42)-net over F₂, using
  - net from sequence [i] based on digital (54, 41)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
      - an explicitly constructive algebraic function field [i]

(236−109, 236, 81)-Net over F₂ — Digital

Digital (127, 236, 81)-net over F₂, using

t-expansion [i] based on digital (126, 236, 81)-net over F₂, using
- net from sequence [i] based on digital (126, 80)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 126 and N(F) ≥ 81, using
    - Drinfeld modules of rank 1 [i]

(236−109, 236, 277)-Net in Base 2 — Upper bound on s

There is no (127, 236, 278)-net in base 2, because

1 times m-reduction [i] would yield (127, 235, 278)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²³⁵, 278, S₂, 108), but
  - the linear programming bound shows that MÂ â‰¥ 8610 120597 997849 103535 796060 969471 925725 360797 699038 214930 323792 633970 252546 589383 983104 / 113493 110947 734375 > 2²³⁵ [i]