MinT - Best Known (74, 74+99, s)-Nets in Base 3

Best Known (74, 74+99, s)-Nets in Base 3

(74, 74+99, 49)-Net over F₃ — Constructive and digital

Digital (74, 173, 49)-net over F₃, using

net from sequence [i] based on digital (74, 48)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 48)-sequence over F₉, using
  - s-reduction based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(74, 74+99, 84)-Net over F₃ — Digital

Digital (74, 173, 84)-net over F₃, using

t-expansion [i] based on digital (71, 173, 84)-net over F₃, using
- net from sequence [i] based on digital (71, 83)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 71 and N(F) ≥ 84, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(74, 74+99, 405)-Net in Base 3 — Upper bound on s

There is no (74, 173, 406)-net in base 3, because

1 times m-reduction [i] would yield (74, 172, 406)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 12286 010758 773595 101831 287334 835109 963939 010257 698958 782262 727553 015026 719434 158285 > 3¹⁷² [i]