MinT - Best Known (133−85, 133, s)-Nets in Base 9

Best Known (133−85, 133, s)-Nets in Base 9

(133−85, 133, 81)-Net over F₉ — Constructive and digital

Digital (48, 133, 81)-net over F₉, using

t-expansion [i] based on digital (32, 133, 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]

(133−85, 133, 163)-Net over F₉ — Digital

Digital (48, 133, 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]

(133−85, 133, 2033)-Net in Base 9 — Upper bound on s

There is no (48, 133, 2034)-net in base 9, because

1 times m-reduction [i] would yield (48, 132, 2034)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 915120 845294 447287 000265 987880 386206 997788 671124 240266 651811 390836 696893 598937 585190 283869 777681 338528 973658 080611 925620 314465 > 9¹³² [i]