MinT - Best Known (129−69, 129, s)-Nets in Base 9

Best Known (129−69, 129, s)-Nets in Base 9

(129−69, 129, 128)-Net over F₉ — Constructive and digital

Digital (60, 129, 128)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (13, 47, 64)-net over F₉, using
  - net from sequence [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]
2. digital (13, 82, 64)-net over F₉, using
  - net from sequence [i] based on digital (13, 63)-sequence over F₉ (see above)

(129−69, 129, 190)-Net over F₉ — Digital

Digital (60, 129, 190)-net over F₉, using

net from sequence [i] based on digital (60, 189)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 60 and N(F) ≥ 190, using
  - Drinfeld modules of rank 1 [i]

(129−69, 129, 6598)-Net in Base 9 — Upper bound on s

There is no (60, 129, 6599)-net in base 9, because

1 times m-reduction [i] would yield (60, 128, 6599)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 139 144552 735374 204826 986846 059899 741256 742645 435737 850146 854799 080458 208796 132807 596177 721198 055644 398527 997987 606202 581617 > 9¹²⁸ [i]