MinT - Best Known (89, 89+31, s)-Nets in Base 2

Best Known (89, 89+31, s)-Nets in Base 2

(89, 89+31, 135)-Net over F₂ — Constructive and digital

Digital (89, 120, 135)-net over F₂, using

trace code for nets [i] based on digital (9, 40, 45)-net over F₈, using
- net from sequence [i] based on digital (9, 44)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using
    - an explicitly constructive algebraic function field [i]

(89, 89+31, 171)-Net over F₂ — Digital

Digital (89, 120, 171)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(89, 89+31, 1548)-Net in Base 2 — Upper bound on s

There is no (89, 120, 1549)-net in base 2, because

1 times m-reduction [i] would yield (89, 119, 1549)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 669219 649643 874762 992666 662358 763576 > 2¹¹⁹ [i]