MinT - Best Known (127−89, 127, s)-Nets in Base 9

Best Known (127−89, 127, s)-Nets in Base 9

(127−89, 127, 81)-Net over F₉ — Constructive and digital

Digital (38, 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]

(127−89, 127, 128)-Net over F₉ — Digital

Digital (38, 127, 128)-net over F₉, using

t-expansion [i] based on digital (33, 127, 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]

(127−89, 127, 1138)-Net in Base 9 — Upper bound on s

There is no (38, 127, 1139)-net in base 9, because

1 times m-reduction [i] would yield (38, 126, 1139)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 1 750330 680205 758438 680272 648224 665011 824660 224698 591570 163692 043053 556398 169891 687954 855679 209215 363874 504190 666708 602145 > 9¹²⁶ [i]