MinT - Best Known (144−89, 144, s)-Nets in Base 9

Best Known (144−89, 144, s)-Nets in Base 9

(144−89, 144, 81)-Net over F₉ — Constructive and digital

Digital (55, 144, 81)-net over F₉, using

t-expansion [i] based on digital (32, 144, 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]

(144−89, 144, 82)-Net in Base 9 — Constructive

(55, 144, 82)-net in base 9, using

base change [i] based on digital (7, 96, 82)-net over F₂₇, using
- net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
    - a function field by SÃ©mirat [i]

(144−89, 144, 182)-Net over F₉ — Digital

Digital (55, 144, 182)-net over F₉, using

t-expansion [i] based on digital (50, 144, 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]

(144−89, 144, 2696)-Net in Base 9 — Upper bound on s

There is no (55, 144, 2697)-net in base 9, because

1 times m-reduction [i] would yield (55, 143, 2697)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 28835 305607 317979 024473 729717 569657 992431 808016 435765 137371 172563 745712 637781 121101 171224 970678 921973 193362 464889 630773 331245 164875 732705 > 9¹⁴³ [i]