MinT - Best Known (69−49, 69, s)-Nets in Base 9

Best Known (69−49, 69, s)-Nets in Base 9

(69−49, 69, 74)-Net over F₉ — Constructive and digital

Digital (20, 69, 74)-net over F₉, using

t-expansion [i] based on digital (17, 69, 74)-net over F₉, using
- net from sequence [i] based on digital (17, 73)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
    - a function field by GarcÃa and Quoos [i]

(69−49, 69, 84)-Net over F₉ — Digital

Digital (20, 69, 84)-net over F₉, using

t-expansion [i] based on digital (19, 69, 84)-net over F₉, using
- net from sequence [i] based on digital (19, 83)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 19 and N(F) ≥ 84, using
    - Drinfeld modules of rank 1 [i]

(69−49, 69, 604)-Net in Base 9 — Upper bound on s

There is no (20, 69, 605)-net in base 9, because

1 times m-reduction [i] would yield (20, 68, 605)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 77402 602790 747036 998000 448627 940228 727758 073494 528582 449411 069121 > 9⁶⁸ [i]