MinT - Best Known (37, 37+69, s)-Nets in Base 9

Best Known (37, 37+69, s)-Nets in Base 9

(37, 37+69, 81)-Net over F₉ — Constructive and digital

Digital (37, 106, 81)-net over F₉, using

t-expansion [i] based on digital (32, 106, 81)-net over F₉, using
- net from sequence [i] based on digital (32, 80)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
    - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(37, 37+69, 128)-Net over F₉ — Digital

Digital (37, 106, 128)-net over F₉, using

t-expansion [i] based on digital (33, 106, 128)-net over F₉, using
- net from sequence [i] based on digital (33, 127)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 33 and N(F) ≥ 128, using
    - a curve from the manYPoints database [i]

(37, 37+69, 1476)-Net in Base 9 — Upper bound on s

There is no (37, 106, 1477)-net in base 9, because

1 times m-reduction [i] would yield (37, 105, 1477)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 15812 500186 577423 313761 511888 664742 073897 029952 005653 685090 573607 920365 647510 672387 153607 127771 578641 > 9¹⁰⁵ [i]