MinT - Best Known (31, 144, s)-Nets in Base 9

Best Known (31, 144, s)-Nets in Base 9

(31, 144, 78)-Net over F₉ — Constructive and digital

Digital (31, 144, 78)-net over F₉, using

t-expansion [i] based on digital (22, 144, 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]

(31, 144, 120)-Net over F₉ — Digital

Digital (31, 144, 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]

(31, 144, 707)-Net in Base 9 — Upper bound on s

There is no (31, 144, 708)-net in base 9, because

1 times m-reduction [i] would yield (31, 143, 708)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 29182 361904 929925 720739 745118 809753 399242 593228 134219 112605 599970 872743 457146 446033 594145 523263 740162 189473 301227 576728 137056 653093 921025 > 9¹⁴³ [i]