MinT - Best Known (163−121, 163, s)-Nets in Base 8

Best Known (163−121, 163, s)-Nets in Base 8

(163−121, 163, 98)-Net over F₈ — Constructive and digital

Digital (42, 163, 98)-net over F₈, using

t-expansion [i] based on digital (37, 163, 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]

(163−121, 163, 129)-Net over F₈ — Digital

Digital (42, 163, 129)-net over F₈, using

t-expansion [i] based on digital (38, 163, 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]

(163−121, 163, 871)-Net in Base 8 — Upper bound on s

There is no (42, 163, 872)-net in base 8, because

1 times m-reduction [i] would yield (42, 162, 872)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 207 326772 622047 545541 571342 527709 960942 033260 100239 668585 871415 428909 898733 661216 423347 876847 139649 580702 513408 775173 311844 824085 029786 737836 100456 > 8¹⁶² [i]