MinT - Best Known (253−107, 253, s)-Nets in Base 2

Best Known (253−107, 253, s)-Nets in Base 2

(253−107, 253, 75)-Net over F₂ — Constructive and digital

Digital (146, 253, 75)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (39, 92, 33)-net over F₂, using
  - net from sequence [i] based on digital (39, 32)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 39 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]
2. digital (54, 161, 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]

(253−107, 253, 99)-Net over F₂ — Digital

Digital (146, 253, 99)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(253−107, 253, 481)-Net in Base 2 — Upper bound on s

There is no (146, 253, 482)-net in base 2, because

1 times m-reduction [i] would yield (146, 252, 482)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 7373 113925 666381 312625 755321 863924 354216 943724 936603 874060 116445 676219 843440 > 2²⁵² [i]