MinT - Best Known (117−75, 117, s)-Nets in Base 9

Best Known (117−75, 117, s)-Nets in Base 9

(117−75, 117, 81)-Net over F₉ — Constructive and digital

Digital (42, 117, 81)-net over F₉, using

t-expansion [i] based on digital (32, 117, 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]

(117−75, 117, 140)-Net over F₉ — Digital

Digital (42, 117, 140)-net over F₉, using

t-expansion [i] based on digital (39, 117, 140)-net over F₉, using
- net from sequence [i] based on digital (39, 139)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 39 and N(F) ≥ 140, using
    - a curve from the manYPoints database [i]

(117−75, 117, 1774)-Net in Base 9 — Upper bound on s

There is no (42, 117, 1775)-net in base 9, because

1 times m-reduction [i] would yield (42, 116, 1775)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 499 705170 940028 777473 184129 038932 237329 506378 622772 478441 672597 240609 610116 446221 417465 832091 647975 228680 263769 > 9¹¹⁶ [i]