MinT - Best Known (126−45, 126, s)-Nets in Base 2

Best Known (126−45, 126, s)-Nets in Base 2

(126−45, 126, 66)-Net over F₂ — Constructive and digital

Digital (81, 126, 66)-net over F₂, using

6 times m-reduction [i] based on digital (81, 132, 66)-net over F₂, using
- trace code for nets [i] based on digital (15, 66, 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]

(126−45, 126, 82)-Net over F₂ — Digital

Digital (81, 126, 82)-net over F₂, using

trace code for nets [i] based on digital (18, 63, 41)-net over F₄, using
- net from sequence [i] based on digital (18, 40)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 18 and N(F) ≥ 41, using
    - Drinfeld modules of rank 1 [i]

(126−45, 126, 433)-Net in Base 2 — Upper bound on s

There is no (81, 126, 434)-net in base 2, because

1 times m-reduction [i] would yield (81, 125, 434)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 44 010205 011145 650380 802166 885599 548112 > 2¹²⁵ [i]