MinT - Best Known (134−97, 134, s)-Nets in Base 9

Best Known (134−97, 134, s)-Nets in Base 9

(134−97, 134, 81)-Net over F₉ — Constructive and digital

Digital (37, 134, 81)-net over F₉, using

t-expansion [i] based on digital (32, 134, 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]

(134−97, 134, 128)-Net over F₉ — Digital

Digital (37, 134, 128)-net over F₉, using

t-expansion [i] based on digital (33, 134, 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]

(134−97, 134, 1002)-Net in Base 9 — Upper bound on s

There is no (37, 134, 1003)-net in base 9, because

1 times m-reduction [i] would yield (37, 133, 1003)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 8 245360 946607 706595 271828 150344 999168 726097 398312 810161 517114 845874 902353 939950 846965 339201 227273 328715 540544 693162 018606 560385 > 9¹³³ [i]