MinT - Best Known (89, 89+137, s)-Nets in Base 3

Best Known (89, 89+137, s)-Nets in Base 3

(89, 89+137, 64)-Net over F₃ — Constructive and digital

Digital (89, 226, 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]

(89, 89+137, 96)-Net over F₃ — Digital

Digital (89, 226, 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]

(89, 89+137, 431)-Net in Base 3 — Upper bound on s

There is no (89, 226, 432)-net in base 3, because

1 times m-reduction [i] would yield (89, 225, 432)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 231496 928408 517084 521892 103982 208489 819528 226914 353518 320638 683216 537486 324653 443540 250706 701846 793860 077697 > 3²²⁵ [i]