MinT - Best Known (111−77, 111, s)-Nets in Base 9

Best Known (111−77, 111, s)-Nets in Base 9

(111−77, 111, 81)-Net over F₉ — Constructive and digital

Digital (34, 111, 81)-net over F₉, using

t-expansion [i] based on digital (32, 111, 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]

(111−77, 111, 128)-Net over F₉ — Digital

Digital (34, 111, 128)-net over F₉, using

t-expansion [i] based on digital (33, 111, 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]

(111−77, 111, 1063)-Net in Base 9 — Upper bound on s

There is no (34, 111, 1064)-net in base 9, because

1 times m-reduction [i] would yield (34, 110, 1064)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 948 889332 551195 937150 815513 713556 455938 661256 810443 659932 627310 876038 767648 935098 102898 159265 274312 065665 > 9¹¹⁰ [i]