MinT - Best Known (144−105, 144, s)-Nets in Base 9

Best Known (144−105, 144, s)-Nets in Base 9

(144−105, 144, 81)-Net over F₉ — Constructive and digital

Digital (39, 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]

(144−105, 144, 140)-Net over F₉ — Digital

Digital (39, 144, 140)-net over F₉, using

net from sequence [i] based on digital (39, 139)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 39 and N(F) ≥ 140, using
  - a curve from the manYPoints database [i]

(144−105, 144, 1032)-Net in Base 9 — Upper bound on s

There is no (39, 144, 1033)-net in base 9, because

1 times m-reduction [i] would yield (39, 143, 1033)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 29534 103761 828726 708517 737674 514198 103582 576440 486645 510635 759409 901725 806873 177231 764814 162318 822556 623286 266932 811231 066285 193361 411105 > 9¹⁴³ [i]