MinT - Best Known (165−123, 165, s)-Nets in Base 8

Best Known (165−123, 165, s)-Nets in Base 8

(165−123, 165, 98)-Net over F₈ — Constructive and digital

Digital (42, 165, 98)-net over F₈, using

t-expansion [i] based on digital (37, 165, 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]

(165−123, 165, 129)-Net over F₈ — Digital

Digital (42, 165, 129)-net over F₈, using

t-expansion [i] based on digital (38, 165, 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]

(165−123, 165, 863)-Net in Base 8 — Upper bound on s

There is no (42, 165, 864)-net in base 8, because

1 times m-reduction [i] would yield (42, 164, 864)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 13168 705401 286048 780729 171521 491398 702643 482526 123072 219502 521163 577534 482571 676149 352815 859319 102539 368647 203723 646269 917018 995321 254165 045089 275998 > 8¹⁶⁴ [i]