MinT - Best Known (137−51, 137, s)-Nets in Base 2

Best Known (137−51, 137, s)-Nets in Base 2

(137−51, 137, 66)-Net over F₂ — Constructive and digital

Digital (86, 137, 66)-net over F₂, using

5 times m-reduction [i] based on digital (86, 142, 66)-net over F₂, using
- trace code for nets [i] based on digital (15, 71, 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]

(137−51, 137, 80)-Net over F₂ — Digital

Digital (86, 137, 80)-net over F₂, using

1 times m-reduction [i] based on digital (86, 138, 80)-net over F₂, using
- trace code for nets [i] based on digital (17, 69, 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]

(137−51, 137, 406)-Net in Base 2 — Upper bound on s

There is no (86, 137, 407)-net in base 2, because

1 times m-reduction [i] would yield (86, 136, 407)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 91387 780079 405481 525602 236251 872388 986208 > 2¹³⁶ [i]