MinT - Best Known (27, 27+31, s)-Nets in Base 9

Best Known (27, 27+31, s)-Nets in Base 9

(27, 27+31, 78)-Net over F₉ — Constructive and digital

Digital (27, 58, 78)-net over F₉, using

t-expansion [i] based on digital (22, 58, 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]

(27, 27+31, 82)-Net in Base 9 — Constructive

(27, 58, 82)-net in base 9, using

2 times m-reduction [i] based on (27, 60, 82)-net in base 9, using
- base change [i] based on digital (7, 40, 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]

(27, 27+31, 110)-Net over F₉ — Digital

Digital (27, 58, 110)-net over F₉, using

t-expansion [i] based on digital (26, 58, 110)-net over F₉, using
- net from sequence [i] based on digital (26, 109)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 26 and N(F) ≥ 110, using
    - a curve from the manYPoints database [i]

(27, 27+31, 3385)-Net in Base 9 — Upper bound on s

There is no (27, 58, 3386)-net in base 9, because

1 times m-reduction [i] would yield (27, 57, 3386)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 2 467137 797702 366020 504293 638693 748335 525056 239547 833585 > 9⁵⁷ [i]