MinT - Best Known (126−85, 126, s)-Nets in Base 8

Best Known (126−85, 126, s)-Nets in Base 8

(126−85, 126, 98)-Net over F₈ — Constructive and digital

Digital (41, 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]

(126−85, 126, 129)-Net over F₈ — Digital

Digital (41, 126, 129)-net over F₈, using

t-expansion [i] based on digital (38, 126, 129)-net over F₈, using
- net from sequence [i] based on digital (38, 128)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 38 and N(F) ≥ 129, using
    - a curve from the manYPoints database [i]

(126−85, 126, 1123)-Net in Base 8 — Upper bound on s

There is no (41, 126, 1124)-net in base 8, because

1 times m-reduction [i] would yield (41, 125, 1124)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 79571 182415 805394 757165 250829 555443 978198 209224 515933 019064 527496 890071 706914 103061 683144 516522 075741 539014 441112 > 8¹²⁵ [i]