MinT - Best Known (141−111, 141, s)-Nets in Base 9

Best Known (141−111, 141, s)-Nets in Base 9

(141−111, 141, 78)-Net over F₉ — Constructive and digital

Digital (30, 141, 78)-net over F₉, using

t-expansion [i] based on digital (22, 141, 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]

(141−111, 141, 110)-Net over F₉ — Digital

Digital (30, 141, 110)-net over F₉, using

t-expansion [i] based on digital (26, 141, 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]

(141−111, 141, 682)-Net in Base 9 — Upper bound on s

There is no (30, 141, 683)-net in base 9, because

1 times m-reduction [i] would yield (30, 140, 683)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 39 844404 891347 032854 109327 428879 509942 554100 593450 930272 577574 221340 941228 244369 705311 044766 649593 360278 381883 982698 913969 505449 512041 > 9¹⁴⁰ [i]