MinT - Best Known (99, 99+43, s)-Nets in Base 9

Best Known (99, 99+43, s)-Nets in Base 9

(99, 99+43, 768)-Net over F₉ — Constructive and digital

Digital (99, 142, 768)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (3, 24, 28)-net over F₉, using
  - net from sequence [i] based on digital (3, 27)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 3 and N(F) ≥ 28, using
      - the Hermitian function field over F₉ [i]
2. digital (75, 118, 740)-net over F₉, using
  - trace code for nets [i] based on digital (16, 59, 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]

(99, 99+43, 3496)-Net over F₉ — Digital

Digital (99, 142, 3496)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(99, 99+43, 2769857)-Net in Base 9 — Upper bound on s

There is no (99, 142, 2769858)-net in base 9, because

1 times m-reduction [i] would yield (99, 141, 2769858)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 353 341778 187548 287862 537851 932223 137687 605624 770730 921015 453930 420343 951143 446444 731907 654393 588413 569916 837797 560947 196356 583100 916945 > 9¹⁴¹ [i]