MinT - Best Known (89, 89+55, s)-Nets in Base 2

Best Known (89, 89+55, s)-Nets in Base 2

(89, 89+55, 66)-Net over F₂ — Constructive and digital

Digital (89, 144, 66)-net over F₂, using

4 times m-reduction [i] based on digital (89, 148, 66)-net over F₂, using
- trace code for nets [i] based on digital (15, 74, 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]

(89, 89+55, 80)-Net over F₂ — Digital

Digital (89, 144, 80)-net over F₂, using

trace code for nets [i] based on digital (17, 72, 40)-net over F₄, using
- net from sequence [i] based on digital (17, 39)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 17 and N(F) ≥ 40, using
    - Drinfeld modules of rank 1 [i]

(89, 89+55, 390)-Net in Base 2 — Upper bound on s

There is no (89, 144, 391)-net in base 2, because

1 times m-reduction [i] would yield (89, 143, 391)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 11 214795 095758 306585 130589 999188 675150 676448 > 2¹⁴³ [i]