MinT - Best Known (51−29, 51, s)-Nets in Base 3

Best Known (51−29, 51, s)-Nets in Base 3

(51−29, 51, 32)-Net over F₃ — Constructive and digital

Digital (22, 51, 32)-net over F₃, using

t-expansion [i] based on digital (21, 51, 32)-net over F₃, using
- net from sequence [i] based on digital (21, 31)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 21 and N(F) ≥ 32, using
    - an explicitly constructive algebraic function field [i]

(51−29, 51, 139)-Net in Base 3 — Upper bound on s

There is no (22, 51, 140)-net in base 3, because

1 times m-reduction [i] would yield (22, 50, 140)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 725415 259159 910278 032777 > 3⁵⁰ [i]