MinT - Best Known (135−81, 135, s)-Nets in Base 9

Best Known (135−81, 135, s)-Nets in Base 9

(135−81, 135, 81)-Net over F₉ — Constructive and digital

Digital (54, 135, 81)-net over F₉, using

t-expansion [i] based on digital (32, 135, 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]

(135−81, 135, 88)-Net in Base 9 — Constructive

(54, 135, 88)-net in base 9, using

base change [i] based on digital (9, 90, 88)-net over F₂₇, using
- net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
    - a function field by SÃ©mirat [i]

(135−81, 135, 182)-Net over F₉ — Digital

Digital (54, 135, 182)-net over F₉, using

t-expansion [i] based on digital (50, 135, 182)-net over F₉, using
- net from sequence [i] based on digital (50, 181)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 50 and N(F) ≥ 182, using
    - a curve from the manYPoints database [i]

(135−81, 135, 3075)-Net in Base 9 — Upper bound on s

There is no (54, 135, 3076)-net in base 9, because

1 times m-reduction [i] would yield (54, 134, 3076)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 73 944558 802552 989562 329335 196505 484890 441755 107767 959037 061823 236624 293207 028824 651794 932329 572114 072442 784354 920613 351172 866817 > 9¹³⁴ [i]