MinT - Best Known (144−45, 144, s)-Nets in Base 9

Best Known (144−45, 144, s)-Nets in Base 9

(144−45, 144, 750)-Net over F₉ — Constructive and digital

Digital (99, 144, 750)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (0, 22, 10)-net over F₉, using
  - net from sequence [i] based on digital (0, 9)-sequence over F₉, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 0 and N(F) ≥ 10, using
      - the rational function field F₉(x) [i]
    - Niederreiter sequence [i]
2. digital (77, 122, 740)-net over F₉, using
  - trace code for nets [i] based on digital (16, 61, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(144−45, 144, 2885)-Net over F₉ — Digital

Digital (99, 144, 2885)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(144−45, 144, 1804413)-Net in Base 9 — Upper bound on s

There is no (99, 144, 1804414)-net in base 9, because

1 times m-reduction [i] would yield (99, 143, 1804414)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 28620 645007 559347 969166 364932 541645 044793 140809 889207 853379 969597 641783 608397 877250 229471 524722 334241 751687 155823 674598 193678 423014 228065 > 9¹⁴³ [i]