MinT - Best Known (126−81, 126, s)-Nets in Base 8

Best Known (126−81, 126, s)-Nets in Base 8

(126−81, 126, 98)-Net over F₈ — Constructive and digital

Digital (45, 126, 98)-net over F₈, using

t-expansion [i] based on digital (37, 126, 98)-net over F₈, using
- net from sequence [i] based on digital (37, 97)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 37 and N(F) ≥ 98, using
    - F₅ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]

(126−81, 126, 144)-Net over F₈ — Digital

Digital (45, 126, 144)-net over F₈, using

net from sequence [i] based on digital (45, 143)-sequence over F₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 45 and N(F) ≥ 144, using
  - Drinfeld modules of rank 1 [i]

(126−81, 126, 1470)-Net in Base 8 — Upper bound on s

There is no (45, 126, 1471)-net in base 8, because

1 times m-reduction [i] would yield (45, 125, 1471)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 77720 147112 984286 715915 787208 471406 214533 051407 081887 874459 049118 809886 375595 972605 170902 984098 187166 630560 869759 > 8¹²⁵ [i]