MinT - Best Known (228−131, 228, s)-Nets in Base 3

Best Known (228−131, 228, s)-Nets in Base 3

(228−131, 228, 64)-Net over F₃ — Constructive and digital

Digital (97, 228, 64)-net over F₃, using

t-expansion [i] based on 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]

(228−131, 228, 96)-Net over F₃ — Digital

Digital (97, 228, 96)-net over F₃, using

t-expansion [i] based on 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]

(228−131, 228, 518)-Net in Base 3 — Upper bound on s

There is no (97, 228, 519)-net in base 3, because

1 times m-reduction [i] would yield (97, 227, 519)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 026703 322731 802303 356163 307346 562131 661483 410561 961810 729372 793343 110042 546534 533089 095497 988101 345269 255055 > 3²²⁷ [i]