MinT - Best Known (71−37, 71, s)-Nets in Base 9

Best Known (71−37, 71, s)-Nets in Base 9

(71−37, 71, 92)-Net over F₉ — Constructive and digital

Digital (34, 71, 92)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (3, 21, 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 (13, 50, 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]

(71−37, 71, 94)-Net in Base 9 — Constructive

(34, 71, 94)-net in base 9, using

1 times m-reduction [i] based on (34, 72, 94)-net in base 9, using
- base change [i] based on digital (10, 48, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]

(71−37, 71, 128)-Net over F₉ — Digital

Digital (34, 71, 128)-net over F₉, using

t-expansion [i] based on digital (33, 71, 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]

(71−37, 71, 4841)-Net in Base 9 — Upper bound on s

There is no (34, 71, 4842)-net in base 9, because

1 times m-reduction [i] would yield (34, 70, 4842)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 6 266855 404125 876711 823880 988949 500358 018405 018226 853771 944907 786081 > 9⁷⁰ [i]