MinT - Best Known (82, 82+33, s)-Nets in Base 9

Best Known (82, 82+33, s)-Nets in Base 9

(82, 82+33, 756)-Net over F₉ — Constructive and digital

Digital (82, 115, 756)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (1, 17, 16)-net over F₉, using
  - net from sequence [i] based on digital (1, 15)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 1 and N(F) ≥ 16, using
      - a function field by GarcÃa and Quoos [i]
2. digital (65, 98, 740)-net over F₉, using
  - trace code for nets [i] based on digital (16, 49, 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]

(82, 82+33, 4312)-Net over F₉ — Digital

Digital (82, 115, 4312)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(82, 82+33, 5350887)-Net in Base 9 — Upper bound on s

There is no (82, 115, 5350888)-net in base 9, because

1 times m-reduction [i] would yield (82, 114, 5350888)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 6 076399 338058 008033 700815 838036 920516 145931 872727 317873 924766 960506 617796 841349 504185 820443 635247 645271 378945 > 9¹¹⁴ [i]