MinT - Best Known (52−27, 52, s)-Nets in Base 3

Best Known (52−27, 52, s)-Nets in Base 3

(52−27, 52, 32)-Net over F₃ — Constructive and digital

Digital (25, 52, 32)-net over F₃, using

t-expansion [i] based on digital (21, 52, 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]

(52−27, 52, 36)-Net over F₃ — Digital

Digital (25, 52, 36)-net over F₃, using

net from sequence [i] based on digital (25, 35)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 25 and N(F) ≥ 36, using
  - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(52−27, 52, 198)-Net in Base 3 — Upper bound on s

There is no (25, 52, 199)-net in base 3, because

1 times m-reduction [i] would yield (25, 51, 199)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 188285 869825 029679 009975 > 3⁵¹ [i]