MinT - Best Known (81−54, 81, s)-Nets in Base 9

Best Known (81−54, 81, s)-Nets in Base 9

(81−54, 81, 78)-Net over F₉ — Constructive and digital

Digital (27, 81, 78)-net over F₉, using

t-expansion [i] based on digital (22, 81, 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]

(81−54, 81, 110)-Net over F₉ — Digital

Digital (27, 81, 110)-net over F₉, using

t-expansion [i] based on digital (26, 81, 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]

(81−54, 81, 979)-Net in Base 9 — Upper bound on s

There is no (27, 81, 980)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 200875 822725 001103 249719 630545 204519 564966 106418 273458 136593 211123 392839 878241 > 9⁸¹ [i]