MinT - Best Known (137−113, 137, s)-Nets in Base 9

Best Known (137−113, 137, s)-Nets in Base 9

(137−113, 137, 78)-Net over F₉ — Constructive and digital

Digital (24, 137, 78)-net over F₉, using

t-expansion [i] based on digital (22, 137, 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]

(137−113, 137, 92)-Net over F₉ — Digital

Digital (24, 137, 92)-net over F₉, using

t-expansion [i] based on digital (23, 137, 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]

(137−113, 137, 529)-Net in Base 9 — Upper bound on s

There is no (24, 137, 530)-net in base 9, because

1 times m-reduction [i] would yield (24, 136, 530)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 6143 811077 603702 345229 287620 816013 932272 226106 168539 029413 597874 539842 905714 781137 576419 535015 686936 233768 330699 384932 315458 873985 > 9¹³⁶ [i]