MinT - Best Known (141−99, 141, s)-Nets in Base 9

Best Known (141−99, 141, s)-Nets in Base 9

(141−99, 141, 81)-Net over F₉ — Constructive and digital

Digital (42, 141, 81)-net over F₉, using

t-expansion [i] based on digital (32, 141, 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]

(141−99, 141, 140)-Net over F₉ — Digital

Digital (42, 141, 140)-net over F₉, using

t-expansion [i] based on digital (39, 141, 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]

(141−99, 141, 1242)-Net in Base 9 — Upper bound on s

There is no (42, 141, 1243)-net in base 9, because

1 times m-reduction [i] would yield (42, 140, 1243)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 40 043633 599473 898019 693368 791334 363835 350467 996160 028140 625414 313853 274394 438474 497061 895232 777186 456530 042021 944314 565547 632421 907289 > 9¹⁴⁰ [i]