MinT - Best Known (138−115, 138, s)-Nets in Base 9

Best Known (138−115, 138, s)-Nets in Base 9

(138−115, 138, 78)-Net over F₉ — Constructive and digital

Digital (23, 138, 78)-net over F₉, using

t-expansion [i] based on digital (22, 138, 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]

(138−115, 138, 92)-Net over F₉ — Digital

Digital (23, 138, 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]

(138−115, 138, 508)-Net in Base 9 — Upper bound on s

There is no (23, 138, 509)-net in base 9, because

1 times m-reduction [i] would yield (23, 137, 509)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 59405 368493 824154 458399 442821 363084 179037 357030 846411 693815 985782 316319 173839 486204 671763 530961 817270 809564 490325 783089 861458 173865 > 9¹³⁷ [i]