MinT - Best Known (251−107, 251, s)-Nets in Base 2

Best Known (251−107, 251, s)-Nets in Base 2

(251−107, 251, 70)-Net over F₂ — Constructive and digital

Digital (144, 251, 70)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (21, 74, 21)-net over F₂, using
  - net from sequence [i] based on digital (21, 20)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 21 and N(F) ≥ 21, using
      - an explicitly constructive algebraic function field [i]
2. digital (70, 177, 49)-net over F₂, using
  - net from sequence [i] based on digital (70, 48)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 69, N(F)Â =Â 48, and 1 place with degree 2 [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]

(251−107, 251, 97)-Net over F₂ — Digital

Digital (144, 251, 97)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(251−107, 251, 467)-Net in Base 2 — Upper bound on s

There is no (144, 251, 468)-net in base 2, because

1 times m-reduction [i] would yield (144, 250, 468)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1888 085978 486497 403952 339333 301550 399596 691821 096008 454270 661475 405925 905952 > 2²⁵⁰ [i]