MinT - Best Known (144−115, 144, s)-Nets in Base 9

Best Known (144−115, 144, s)-Nets in Base 9

(144−115, 144, 78)-Net over F₉ — Constructive and digital

Digital (29, 144, 78)-net over F₉, using

t-expansion [i] based on digital (22, 144, 78)-net over F₉, using
- net from sequence [i] based on digital (22, 77)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
    - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(144−115, 144, 110)-Net over F₉ — Digital

Digital (29, 144, 110)-net over F₉, using

t-expansion [i] based on digital (26, 144, 110)-net over F₉, using
- net from sequence [i] based on digital (26, 109)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 26 and N(F) ≥ 110, using
    - a curve from the manYPoints database [i]

(144−115, 144, 649)-Net in Base 9 — Upper bound on s

There is no (29, 144, 650)-net in base 9, because

1 times m-reduction [i] would yield (29, 143, 650)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 30807 014538 095337 436191 001400 917624 090432 620625 273246 272005 079494 535372 949466 627259 277151 249009 813202 814609 815120 785451 177835 364076 355281 > 9¹⁴³ [i]