MinT - Best Known (126−75, 126, s)-Nets in Base 8

Best Known (126−75, 126, s)-Nets in Base 8

(126−75, 126, 98)-Net over F₈ — Constructive and digital

Digital (51, 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]

(126−75, 126, 144)-Net over F₈ — Digital

Digital (51, 126, 144)-net over F₈, using

t-expansion [i] based on 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]

(126−75, 126, 2330)-Net in Base 8 — Upper bound on s

There is no (51, 126, 2331)-net in base 8, because

1 times m-reduction [i] would yield (51, 125, 2331)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 77303 878229 312249 134137 659179 958795 018839 898139 819870 628398 405121 107844 371855 432528 586456 942846 276421 813503 819616 > 8¹²⁵ [i]