MinT - Best Known (133−109, 133, s)-Nets in Base 9

Best Known (133−109, 133, s)-Nets in Base 9

(133−109, 133, 78)-Net over F₉ — Constructive and digital

Digital (24, 133, 78)-net over F₉, using

t-expansion [i] based on digital (22, 133, 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]

(133−109, 133, 92)-Net over F₉ — Digital

Digital (24, 133, 92)-net over F₉, using

t-expansion [i] based on digital (23, 133, 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]

(133−109, 133, 531)-Net in Base 9 — Upper bound on s

There is no (24, 133, 532)-net in base 9, because

1 times m-reduction [i] would yield (24, 132, 532)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 1 003114 721369 865888 545214 815141 980444 403438 214231 656146 053450 838758 229754 190833 207290 817405 412490 752462 009098 597884 602498 853185 > 9¹³² [i]