MinT - Best Known (69, 69+39, s)-Nets in Base 2

Best Known (69, 69+39, s)-Nets in Base 2

(69, 69+39, 66)-Net over F₂ — Constructive and digital

Digital (69, 108, 66)-net over F₂, using

trace code for nets [i] based on digital (15, 54, 33)-net over F₄, using
- net from sequence [i] based on digital (15, 32)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
    - an explicitly constructive algebraic function field [i]

(69, 69+39, 71)-Net over F₂ — Digital

Digital (69, 108, 71)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(69, 69+39, 365)-Net in Base 2 — Upper bound on s

There is no (69, 108, 366)-net in base 2, because

1 times m-reduction [i] would yield (69, 107, 366)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 162 567067 287971 593258 576066 029668 > 2¹⁰⁷ [i]