MinT - Best Known (126−89, 126, s)-Nets in Base 9

Best Known (126−89, 126, s)-Nets in Base 9

(126−89, 126, 81)-Net over F₉ — Constructive and digital

Digital (37, 126, 81)-net over F₉, using

t-expansion [i] based on digital (32, 126, 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]

(126−89, 126, 128)-Net over F₉ — Digital

Digital (37, 126, 128)-net over F₉, using

t-expansion [i] based on digital (33, 126, 128)-net over F₉, using
- net from sequence [i] based on digital (33, 127)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 33 and N(F) ≥ 128, using
    - a curve from the manYPoints database [i]

(126−89, 126, 1081)-Net in Base 9 — Upper bound on s

There is no (37, 126, 1082)-net in base 9, because

1 times m-reduction [i] would yield (37, 125, 1082)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 192778 930289 292684 772266 320799 234858 737541 030078 939072 938364 925996 919431 827211 234212 441507 347760 754915 160499 896627 735361 > 9¹²⁵ [i]