MinT - Best Known (172−97, 172, s)-Nets in Base 3

Best Known (172−97, 172, s)-Nets in Base 3

(172−97, 172, 50)-Net over F₃ — Constructive and digital

Digital (75, 172, 50)-net over F₃, using

net from sequence [i] based on digital (75, 49)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 49)-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]

(172−97, 172, 84)-Net over F₃ — Digital

Digital (75, 172, 84)-net over F₃, using

t-expansion [i] based on digital (71, 172, 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]

(172−97, 172, 423)-Net in Base 3 — Upper bound on s

There is no (75, 172, 424)-net in base 3, because

1 times m-reduction [i] would yield (75, 171, 424)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3967 654478 742610 100511 048445 066583 382220 812781 299321 394443 152821 422295 024255 130625 > 3¹⁷¹ [i]