MinT - Best Known (172−131, 172, s)-Nets in Base 8

Best Known (172−131, 172, s)-Nets in Base 8

(172−131, 172, 98)-Net over F₈ — Constructive and digital

Digital (41, 172, 98)-net over F₈, using

t-expansion [i] based on digital (37, 172, 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]

(172−131, 172, 129)-Net over F₈ — Digital

Digital (41, 172, 129)-net over F₈, using

t-expansion [i] based on digital (38, 172, 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]

(172−131, 172, 809)-Net in Base 8 — Upper bound on s

There is no (41, 172, 810)-net in base 8, because

1 times m-reduction [i] would yield (41, 171, 810)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 28199 236530 895554 473076 848063 392245 224066 478552 937190 264688 983577 727463 036031 624992 960287 859606 674788 165549 307764 014662 417758 295708 679410 898229 092408 555812 > 8¹⁷¹ [i]