MinT - Best Known (170−103, 170, s)-Nets in Base 8

Best Known (170−103, 170, s)-Nets in Base 8

(170−103, 170, 98)-Net over F₈ — Constructive and digital

Digital (67, 170, 98)-net over F₈, using

t-expansion [i] based on digital (37, 170, 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]

(170−103, 170, 144)-Net over F₈ — Digital

Digital (67, 170, 144)-net over F₈, using

t-expansion [i] based on digital (45, 170, 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]

(170−103, 170, 2756)-Net in Base 8 — Upper bound on s

There is no (67, 170, 2757)-net in base 8, because

1 times m-reduction [i] would yield (67, 169, 2757)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 425 208029 080651 072127 754083 026274 359065 088862 460847 775176 080768 170809 502089 339824 381439 381433 431720 728915 513925 008709 967026 087020 710839 587765 508965 921200 > 8¹⁶⁹ [i]