MinT - Best Known (143−113, 143, s)-Nets in Base 9

Best Known (143−113, 143, s)-Nets in Base 9

(143−113, 143, 78)-Net over F₉ — Constructive and digital

Digital (30, 143, 78)-net over F₉, using

t-expansion [i] based on digital (22, 143, 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]

(143−113, 143, 110)-Net over F₉ — Digital

Digital (30, 143, 110)-net over F₉, using

t-expansion [i] based on digital (26, 143, 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]

(143−113, 143, 679)-Net in Base 9 — Upper bound on s

There is no (30, 143, 680)-net in base 9, because

1 times m-reduction [i] would yield (30, 142, 680)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 3381 046520 837767 330256 444954 975721 572773 848206 675608 234045 998420 120376 332565 633880 110168 944205 129233 881389 045226 606363 494802 033362 459137 > 9¹⁴² [i]