MinT - Best Known (116−85, 116, s)-Nets in Base 9

Best Known (116−85, 116, s)-Nets in Base 9

(116−85, 116, 78)-Net over F₉ — Constructive and digital

Digital (31, 116, 78)-net over F₉, using

t-expansion [i] based on digital (22, 116, 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]

(116−85, 116, 120)-Net over F₉ — Digital

Digital (31, 116, 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]

(116−85, 116, 820)-Net in Base 9 — Upper bound on s

There is no (31, 116, 821)-net in base 9, because

1 times m-reduction [i] would yield (31, 115, 821)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 55 325821 752786 494589 907735 247749 626565 419520 349587 063496 313327 510243 042484 514595 798063 464327 044300 767723 838545 > 9¹¹⁵ [i]