MinT - Best Known (168−119, 168, s)-Nets in Base 8

Best Known (168−119, 168, s)-Nets in Base 8

(168−119, 168, 98)-Net over F₈ — Constructive and digital

Digital (49, 168, 98)-net over F₈, using

t-expansion [i] based on digital (37, 168, 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]

(168−119, 168, 144)-Net over F₈ — Digital

Digital (49, 168, 144)-net over F₈, using

t-expansion [i] based on digital (45, 168, 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]

(168−119, 168, 1136)-Net in Base 8 — Upper bound on s

There is no (49, 168, 1137)-net in base 8, because

1 times m-reduction [i] would yield (49, 167, 1137)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6 772497 309926 826822 559525 754881 589083 890705 555155 302856 625355 628956 553248 213528 478951 052835 149739 264850 051634 705074 583301 008799 315172 967677 459795 733056 > 8¹⁶⁷ [i]