MinT - Best Known (72−39, 72, s)-Nets in Base 3

Best Known (72−39, 72, s)-Nets in Base 3

(72−39, 72, 38)-Net over F₃ — Constructive and digital

Digital (33, 72, 38)-net over F₃, using

t-expansion [i] based on digital (32, 72, 38)-net over F₃, using
- net from sequence [i] based on digital (32, 37)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 32 and N(F) ≥ 38, using
    - an explicitly constructive algebraic function field [i]

(72−39, 72, 46)-Net over F₃ — Digital

Digital (33, 72, 46)-net over F₃, using

net from sequence [i] based on digital (33, 45)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 33 and N(F) ≥ 46, using
  - a curve from the manYPoints database [i]

(72−39, 72, 222)-Net in Base 3 — Upper bound on s

There is no (33, 72, 223)-net in base 3, because

1 times m-reduction [i] would yield (33, 71, 223)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 7768 933858 034468 370780 487144 471899 > 3⁷¹ [i]