MinT - Best Known (144−81, 144, s)-Nets in Base 2

Best Known (144−81, 144, s)-Nets in Base 2

(144−81, 144, 43)-Net over F₂ — Constructive and digital

Digital (63, 144, 43)-net over F₂, using

t-expansion [i] based on digital (59, 144, 43)-net over F₂, using
- net from sequence [i] based on digital (59, 42)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 54, N(F)Â =Â 42, and 1 place with degree 6 [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]

(144−81, 144, 44)-Net over F₂ — Digital

Digital (63, 144, 44)-net over F₂, using

t-expansion [i] based on digital (62, 144, 44)-net over F₂, using
- net from sequence [i] based on digital (62, 43)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 62 and N(F) ≥ 44, using
    - a function field from a table by Niederreiter and Xing [i]

(144−81, 144, 135)-Net in Base 2 — Upper bound on s

There is no (63, 144, 136)-net in base 2, because

1 times m-reduction [i] would yield (63, 143, 136)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 11 675013 985422 869002 303393 192834 173151 672976 > 2¹⁴³ [i]