MinT - Best Known (233−137, 233, s)-Nets in Base 3

Best Known (233−137, 233, s)-Nets in Base 3

(233−137, 233, 64)-Net over F₃ — Constructive and digital

Digital (96, 233, 64)-net over F₃, using

t-expansion [i] based on digital (89, 233, 64)-net over F₃, using
- net from sequence [i] based on digital (89, 63)-sequence over F₃, using
  - base reduction for sequences [i] 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]

(233−137, 233, 96)-Net over F₃ — Digital

Digital (96, 233, 96)-net over F₃, using

t-expansion [i] based on digital (89, 233, 96)-net over F₃, using
- net from sequence [i] based on digital (89, 95)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 89 and N(F) ≥ 96, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(233−137, 233, 490)-Net in Base 3 — Upper bound on s

There is no (96, 233, 491)-net in base 3, because

1 times m-reduction [i] would yield (96, 232, 491)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 504 841358 507882 304929 298275 193140 874582 177539 209202 796180 912212 737773 611344 448100 699349 066291 820270 367739 557465 > 3²³² [i]