MinT - Best Known (149−85, 149, s)-Nets in Base 9

Best Known (149−85, 149, s)-Nets in Base 9

(149−85, 149, 165)-Net over F₉ — Constructive and digital

Digital (64, 149, 165)-net over F₉, using

net from sequence [i] based on digital (64, 164)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 64 and N(F) ≥ 165, using
  - T₄ from the second tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(149−85, 149, 192)-Net over F₉ — Digital

Digital (64, 149, 192)-net over F₉, using

t-expansion [i] based on digital (61, 149, 192)-net over F₉, using
- net from sequence [i] based on digital (61, 191)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 61 and N(F) ≥ 192, using
    - Drinfeld modules of rank 1 [i]

(149−85, 149, 4730)-Net in Base 9 — Upper bound on s

There is no (64, 149, 4731)-net in base 9, because

1 times m-reduction [i] would yield (64, 148, 4731)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 1694 253845 210544 719522 015105 008260 780764 769230 988415 042337 231448 527490 476687 457033 726231 878743 725554 526530 158310 772637 114264 168880 916811 504113 > 9¹⁴⁸ [i]