MinT - Best Known (71−40, 71, s)-Nets in Base 9

Best Known (71−40, 71, s)-Nets in Base 9

(71−40, 71, 78)-Net over F₉ — Constructive and digital

Digital (31, 71, 78)-net over F₉, using

t-expansion [i] based on digital (22, 71, 78)-net over F₉, using
- net from sequence [i] based on digital (22, 77)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
    - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(71−40, 71, 82)-Net in Base 9 — Constructive

(31, 71, 82)-net in base 9, using

1 times m-reduction [i] based on (31, 72, 82)-net in base 9, using
- base change [i] based on digital (7, 48, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]

(71−40, 71, 120)-Net over F₉ — Digital

Digital (31, 71, 120)-net over F₉, using

net from sequence [i] based on digital (31, 119)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 31 and N(F) ≥ 120, using
  - a curve from the manYPoints database [i]

(71−40, 71, 2521)-Net in Base 9 — Upper bound on s

There is no (31, 71, 2522)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 56 461031 592536 877915 030328 832413 948484 883789 925065 607479 853699 279041 > 9⁷¹ [i]