MinT - Best Known (39, 39+61, s)-Nets in Base 3

Best Known (39, 39+61, s)-Nets in Base 3

(39, 39+61, 42)-Net over F₃ — Constructive and digital

Digital (39, 100, 42)-net over F₃, using

net from sequence [i] based on digital (39, 41)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 39 and N(F) ≥ 42, using
  - an explicitly constructive algebraic function field [i]

(39, 39+61, 52)-Net over F₃ — Digital

Digital (39, 100, 52)-net over F₃, using

t-expansion [i] based on digital (37, 100, 52)-net over F₃, using
- net from sequence [i] based on digital (37, 51)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 37 and N(F) ≥ 52, using
    - a curve from the manYPoints database [i]

(39, 39+61, 198)-Net in Base 3 — Upper bound on s

There is no (39, 100, 199)-net in base 3, because

1 times m-reduction [i] would yield (39, 99, 199)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 194172 948945 344479 929376 478207 912588 647885 890837 > 3⁹⁹ [i]