MinT - Best Known (90, 90+135, s)-Nets in Base 3

Best Known (90, 90+135, s)-Nets in Base 3

(90, 90+135, 64)-Net over F₃ — Constructive and digital

Digital (90, 225, 64)-net over F₃, using

t-expansion [i] based on digital (89, 225, 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]

(90, 90+135, 96)-Net over F₃ — Digital

Digital (90, 225, 96)-net over F₃, using

t-expansion [i] based on digital (89, 225, 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]

(90, 90+135, 444)-Net in Base 3 — Upper bound on s

There is no (90, 225, 445)-net in base 3, because

1 times m-reduction [i] would yield (90, 224, 445)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 80635 371218 222670 135517 045475 184533 519278 647685 856879 916914 022907 593828 623223 586690 059825 399341 001022 534155 > 3²²⁴ [i]