MinT - Best Known (45, 45+121, s)-Nets in Base 8

Best Known (45, 45+121, s)-Nets in Base 8

(45, 45+121, 98)-Net over F₈ — Constructive and digital

Digital (45, 166, 98)-net over F₈, using

t-expansion [i] based on digital (37, 166, 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]

(45, 45+121, 144)-Net over F₈ — Digital

Digital (45, 166, 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]

(45, 45+121, 970)-Net in Base 8 — Upper bound on s

There is no (45, 166, 971)-net in base 8, because

1 times m-reduction [i] would yield (45, 165, 971)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 102319 491824 312567 692302 664425 812937 161769 185776 102188 591193 753558 424052 292715 977540 214400 807214 254719 494574 538534 532303 636254 061402 265604 793370 705872 > 8¹⁶⁵ [i]