MinT - Best Known (140−117, 140, s)-Nets in Base 9

Best Known (140−117, 140, s)-Nets in Base 9

(140−117, 140, 78)-Net over F₉ — Constructive and digital

Digital (23, 140, 78)-net over F₉, using

t-expansion [i] based on digital (22, 140, 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]

(140−117, 140, 92)-Net over F₉ — Digital

Digital (23, 140, 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]

(140−117, 140, 508)-Net in Base 9 — Upper bound on s

There is no (23, 140, 509)-net in base 9, because

1 times m-reduction [i] would yield (23, 139, 509)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 4 749029 239307 218730 959990 806335 080120 937368 353148 755640 321528 801105 498612 903198 798588 290049 627327 838993 175621 801046 589924 125100 548689 > 9¹³⁹ [i]