MinT - Best Known (247−105, 247, s)-Nets in Base 2

Best Known (247−105, 247, s)-Nets in Base 2

(247−105, 247, 69)-Net over F₂ — Constructive and digital

Digital (142, 247, 69)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (39, 91, 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 (51, 156, 36)-net over F₂, using
  - net from sequence [i] based on digital (51, 35)-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 3 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]

(247−105, 247, 96)-Net over F₂ — Digital

Digital (142, 247, 96)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(247−105, 247, 464)-Net in Base 2 — Upper bound on s

There is no (142, 247, 465)-net in base 2, because

1 times m-reduction [i] would yield (142, 246, 465)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 121 318073 051747 276832 957325 630851 926420 508764 167276 621859 416629 171405 146864 > 2²⁴⁶ [i]