MinT - Best Known (127−83, 127, s)-Nets in Base 9

Best Known (127−83, 127, s)-Nets in Base 9

(127−83, 127, 81)-Net over F₉ — Constructive and digital

Digital (44, 127, 81)-net over F₉, using

t-expansion [i] based on digital (32, 127, 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]

(127−83, 127, 147)-Net over F₉ — Digital

Digital (44, 127, 147)-net over F₉, using

t-expansion [i] based on digital (43, 127, 147)-net over F₉, using
- net from sequence [i] based on digital (43, 146)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 43 and N(F) ≥ 147, using
    - a curve from the manYPoints database [i]

(127−83, 127, 1702)-Net in Base 9 — Upper bound on s

There is no (44, 127, 1703)-net in base 9, because

1 times m-reduction [i] would yield (44, 126, 1703)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 1 744790 133950 594803 810836 066232 465368 977623 349202 037232 720144 897550 668892 775851 588958 725915 972919 032161 677098 382334 935545 > 9¹²⁶ [i]