MinT - Best Known (145−122, 145, s)-Nets in Base 9

Best Known (145−122, 145, s)-Nets in Base 9

(145−122, 145, 78)-Net over F₉ — Constructive and digital

Digital (23, 145, 78)-net over F₉, using

t-expansion [i] based on digital (22, 145, 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]

(145−122, 145, 92)-Net over F₉ — Digital

Digital (23, 145, 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]

(145−122, 145, 508)-Net in Base 9 — Upper bound on s

There is no (23, 145, 509)-net in base 9, because

2 times m-reduction [i] would yield (23, 143, 509)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 29024 298757 528591 171990 129589 678557 862956 169729 219612 110071 289217 782589 435770 931825 724423 194322 120219 156920 879620 614091 827106 074129 404385 > 9¹⁴³ [i]