MinT - Best Known (137−89, 137, s)-Nets in Base 9

Best Known (137−89, 137, s)-Nets in Base 9

(137−89, 137, 81)-Net over F₉ — Constructive and digital

Digital (48, 137, 81)-net over F₉, using

t-expansion [i] based on digital (32, 137, 81)-net over F₉, using
- net from sequence [i] based on digital (32, 80)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
    - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(137−89, 137, 163)-Net over F₉ — Digital

Digital (48, 137, 163)-net over F₉, using

net from sequence [i] based on digital (48, 162)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 48 and N(F) ≥ 163, using
  - Drinfeld modules of rank 1 [i]

(137−89, 137, 1893)-Net in Base 9 — Upper bound on s

There is no (48, 137, 1894)-net in base 9, because

1 times m-reduction [i] would yield (48, 136, 1894)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 6101 882974 127615 890359 981935 599587 910594 918026 536533 625207 585937 757899 876081 647718 450813 370546 223878 277355 547034 351748 844461 730497 > 9¹³⁶ [i]