MinT - Best Known (68−45, 68, s)-Nets in Base 9

Best Known (68−45, 68, s)-Nets in Base 9

(68−45, 68, 78)-Net over F₉ — Constructive and digital

Digital (23, 68, 78)-net over F₉, using

t-expansion [i] based on digital (22, 68, 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]

(68−45, 68, 92)-Net over F₉ — Digital

Digital (23, 68, 92)-net over F₉, using

net from sequence [i] based on digital (23, 91)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 23 and N(F) ≥ 92, using
  - Drinfeld modules of rank 1 [i]

(68−45, 68, 898)-Net in Base 9 — Upper bound on s

There is no (23, 68, 899)-net in base 9, because

1 times m-reduction [i] would yield (23, 67, 899)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 8691 957758 613009 784765 197294 431718 825939 212744 005660 715038 561425 > 9⁶⁷ [i]