MinT - Best Known (72−27, 72, s)-Nets in Base 2

Best Known (72−27, 72, s)-Nets in Base 2

(72−27, 72, 44)-Net over F₂ — Constructive and digital

Digital (45, 72, 44)-net over F₂, using

trace code for nets [i] based on digital (9, 36, 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]

(72−27, 72, 52)-Net over F₂ — Digital

Digital (45, 72, 52)-net over F₂, using

trace code for nets [i] based on digital (9, 36, 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]

(72−27, 72, 231)-Net in Base 2 — Upper bound on s

There is no (45, 72, 232)-net in base 2, because

1 times m-reduction [i] would yield (45, 71, 232)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 2440 019198 984457 797700 > 2⁷¹ [i]