MinT - Best Known (109−85, 109, s)-Nets in Base 9

Best Known (109−85, 109, s)-Nets in Base 9

(109−85, 109, 78)-Net over F₉ — Constructive and digital

Digital (24, 109, 78)-net over F₉, using

t-expansion [i] based on digital (22, 109, 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]

(109−85, 109, 92)-Net over F₉ — Digital

Digital (24, 109, 92)-net over F₉, using

t-expansion [i] based on digital (23, 109, 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]

(109−85, 109, 561)-Net in Base 9 — Upper bound on s

There is no (24, 109, 562)-net in base 9, because

1 times m-reduction [i] would yield (24, 108, 562)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 11 969786 000951 683587 349463 039319 169660 112441 017648 053320 760896 017422 904778 162383 262424 171225 955682 717025 > 9¹⁰⁸ [i]