MinT - Best Known (250−99, 250, s)-Nets in Base 2

Best Known (250−99, 250, s)-Nets in Base 2

(250−99, 250, 77)-Net over F₂ — Constructive and digital

Digital (151, 250, 77)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (48, 97, 35)-net over F₂, using
  - net from sequence [i] based on digital (48, 34)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 2 places with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]
2. digital (54, 153, 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]

(250−99, 250, 114)-Net over F₂ — Digital

Digital (151, 250, 114)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(250−99, 250, 577)-Net in Base 2 — Upper bound on s

There is no (151, 250, 578)-net in base 2, because

1 times m-reduction [i] would yield (151, 249, 578)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 922 539145 452132 895033 800166 498112 830460 123955 939356 368568 165857 678340 867700 > 2²⁴⁹ [i]