MinT - Best Known (37, 82, s)-Nets in Base 9

Best Known (37, 82, s)-Nets in Base 9

(37, 82, 84)-Net over F₉ — Constructive and digital

Digital (37, 82, 84)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (2, 24, 20)-net over F₉, using
  - net from sequence [i] based on digital (2, 19)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 2 and N(F) ≥ 20, using
      - a function field by GarcÃa and Quoos [i]
2. digital (13, 58, 64)-net over F₉, using
  - net from sequence [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]

(37, 82, 88)-Net in Base 9 — Constructive

(37, 82, 88)-net in base 9, using

2 times m-reduction [i] based on (37, 84, 88)-net in base 9, using
- base change [i] based on digital (9, 56, 88)-net over F₂₇, using
  - net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
      - a function field by SÃ©mirat [i]

(37, 82, 128)-Net over F₉ — Digital

Digital (37, 82, 128)-net over F₉, using

t-expansion [i] based on digital (33, 82, 128)-net over F₉, using
- net from sequence [i] based on digital (33, 127)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 33 and N(F) ≥ 128, using
    - a curve from the manYPoints database [i]

(37, 82, 3677)-Net in Base 9 — Upper bound on s

There is no (37, 82, 3678)-net in base 9, because

1 times m-reduction [i] would yield (37, 81, 3678)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 197238 822251 203589 962277 564643 984195 287507 433892 417752 148404 610554 480813 334113 > 9⁸¹ [i]