MinT - Best Known (56−29, 56, s)-Nets in Base 9

Best Known (56−29, 56, s)-Nets in Base 9

(56−29, 56, 78)-Net over F₉ — Constructive and digital

Digital (27, 56, 78)-net over F₉, using

t-expansion [i] based on digital (22, 56, 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]

(56−29, 56, 84)-Net in Base 9 — Constructive

(27, 56, 84)-net in base 9, using

1 times m-reduction [i] based on (27, 57, 84)-net in base 9, using
- base change [i] based on digital (8, 38, 84)-net over F₂₇, using
  - net from sequence [i] based on digital (8, 83)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 8 and N(F) ≥ 84, using
      - a function field by SÃ©mirat [i]

(56−29, 56, 110)-Net over F₉ — Digital

Digital (27, 56, 110)-net over F₉, using

t-expansion [i] based on digital (26, 56, 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]

(56−29, 56, 4229)-Net in Base 9 — Upper bound on s

There is no (27, 56, 4230)-net in base 9, because

1 times m-reduction [i] would yield (27, 55, 4230)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 30440 822225 899865 865280 242383 447887 658959 719684 450401 > 9⁵⁵ [i]