MinT - Best Known (69−25, 69, s)-Nets in Base 2

Best Known (69−25, 69, s)-Nets in Base 2

(69−25, 69, 44)-Net over F₂ — Constructive and digital

Digital (44, 69, 44)-net over F₂, using

1 times m-reduction [i] based on digital (44, 70, 44)-net over F₂, using
- trace code for nets [i] based on digital (9, 35, 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]

(69−25, 69, 52)-Net over F₂ — Digital

Digital (44, 69, 52)-net over F₂, using

1 times m-reduction [i] based on digital (44, 70, 52)-net over F₂, using
- trace code for nets [i] based on digital (9, 35, 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]

(69−25, 69, 251)-Net in Base 2 — Upper bound on s

There is no (44, 69, 252)-net in base 2, because

1 times m-reduction [i] would yield (44, 68, 252)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 299 761361 620065 180572 > 2⁶⁸ [i]