MinT - Best Known (251−143, 251, s)-Nets in Base 2

Best Known (251−143, 251, s)-Nets in Base 2

(251−143, 251, 56)-Net over F₂ — Constructive and digital

Digital (108, 251, 56)-net over F₂, using

t-expansion [i] based on digital (105, 251, 56)-net over F₂, using
- net from sequence [i] based on digital (105, 55)-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, 1 place with degree 2, and 7 places with degree 6 [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]

(251−143, 251, 65)-Net over F₂ — Digital

Digital (108, 251, 65)-net over F₂, using

t-expansion [i] based on digital (95, 251, 65)-net over F₂, using
- net from sequence [i] based on digital (95, 64)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 95 and N(F) ≥ 65, using
    - Drinfeld modules of rank 1 [i]

(251−143, 251, 220)-Net in Base 2 — Upper bound on s

There is no (108, 251, 221)-net in base 2, because

1 times m-reduction [i] would yield (108, 250, 221)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1827 034693 164109 605838 569289 059685 987756 968038 015062 732597 061596 228135 092248 > 2²⁵⁰ [i]