MinT - Best Known (228−139, 228, s)-Nets in Base 3

Best Known (228−139, 228, s)-Nets in Base 3

(228−139, 228, 64)-Net over F₃ — Constructive and digital

Digital (89, 228, 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]

(228−139, 228, 96)-Net over F₃ — Digital

Digital (89, 228, 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]

(228−139, 228, 427)-Net in Base 3 — Upper bound on s

There is no (89, 228, 428)-net in base 3, because

1 times m-reduction [i] would yield (89, 227, 428)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 151678 204823 712711 493613 576534 500948 225208 608771 252506 817471 816232 124206 847381 762054 374485 344379 369102 019641 > 3²²⁷ [i]