MinT - Best Known (89−35, 89, s)-Nets in Base 2

Best Known (89−35, 89, s)-Nets in Base 2

(89−35, 89, 44)-Net over F₂ — Constructive and digital

Digital (54, 89, 44)-net over F₂, using

1 times m-reduction [i] based on digital (54, 90, 44)-net over F₂, using
- trace code for nets [i] based on digital (9, 45, 22)-net over F₄, using
  - net from sequence [i] based on digital (9, 21)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 9 and N(F) ≥ 22, using
      - an explicitly constructive algebraic function field [i]

(89−35, 89, 52)-Net over F₂ — Digital

Digital (54, 89, 52)-net over F₂, using

1 times m-reduction [i] based on digital (54, 90, 52)-net over F₂, using
- trace code for nets [i] based on digital (9, 45, 26)-net over F₄, using
  - net from sequence [i] based on digital (9, 25)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 9 and N(F) ≥ 26, using
      - Drinfeld modules of rank 1 [i]

(89−35, 89, 235)-Net in Base 2 — Upper bound on s

There is no (54, 89, 236)-net in base 2, because

1 times m-reduction [i] would yield (54, 88, 236)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 317 998891 475341 429660 299302 > 2⁸⁸ [i]