MinT - Best Known (127−85, 127, s)-Nets in Base 9

Best Known (127−85, 127, s)-Nets in Base 9

(127−85, 127, 81)-Net over F₉ — Constructive and digital

Digital (42, 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]

(127−85, 127, 140)-Net over F₉ — Digital

Digital (42, 127, 140)-net over F₉, using

t-expansion [i] based on digital (39, 127, 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]

(127−85, 127, 1479)-Net in Base 9 — Upper bound on s

There is no (42, 127, 1480)-net in base 9, because

1 times m-reduction [i] would yield (42, 126, 1480)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 1 762654 631703 413516 103500 001719 334056 404384 067629 468535 301513 948498 410020 391507 769735 207368 094005 379891 850463 412058 387329 > 9¹²⁶ [i]