MinT - Best Known (144−87, 144, s)-Nets in Base 9

Best Known (144−87, 144, s)-Nets in Base 9

(144−87, 144, 81)-Net over F₉ — Constructive and digital

Digital (57, 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]

(144−87, 144, 88)-Net in Base 9 — Constructive

(57, 144, 88)-net in base 9, using

base change [i] based on digital (9, 96, 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]

(144−87, 144, 182)-Net over F₉ — Digital

Digital (57, 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]

(144−87, 144, 3119)-Net in Base 9 — Upper bound on s

There is no (57, 144, 3120)-net in base 9, because

1 times m-reduction [i] would yield (57, 143, 3120)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 28654 313569 762164 402909 942669 374470 628593 161123 463840 957235 701739 934652 864193 701373 567082 192879 263163 147717 412675 323197 798159 542096 562817 > 9¹⁴³ [i]