MinT - Best Known (78−45, 78, s)-Nets in Base 9

Best Known (78−45, 78, s)-Nets in Base 9

(78−45, 78, 81)-Net over F₉ — Constructive and digital

Digital (33, 78, 81)-net over F₉, using

t-expansion [i] based on digital (32, 78, 81)-net over F₉, using
- net from sequence [i] based on digital (32, 80)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
    - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(78−45, 78, 82)-Net in Base 9 — Constructive

(33, 78, 82)-net in base 9, using

base change [i] based on digital (7, 52, 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]

(78−45, 78, 128)-Net over F₉ — Digital

Digital (33, 78, 128)-net over F₉, using

net from sequence [i] based on digital (33, 127)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 33 and N(F) ≥ 128, using
  - a curve from the manYPoints database [i]

(78−45, 78, 2461)-Net in Base 9 — Upper bound on s

There is no (33, 78, 2462)-net in base 9, because

1 times m-reduction [i] would yield (33, 77, 2462)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 29 974418 800345 392850 214124 300159 674981 130907 585115 437357 474647 614937 502305 > 9⁷⁷ [i]