MinT - Best Known (42, 157, s)-Nets in Base 8

Best Known (42, 157, s)-Nets in Base 8

(42, 157, 98)-Net over F₈ — Constructive and digital

Digital (42, 157, 98)-net over F₈, using

t-expansion [i] based on digital (37, 157, 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]

(42, 157, 129)-Net over F₈ — Digital

Digital (42, 157, 129)-net over F₈, using

t-expansion [i] based on digital (38, 157, 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]

(42, 157, 898)-Net in Base 8 — Upper bound on s

There is no (42, 157, 899)-net in base 8, because

1 times m-reduction [i] would yield (42, 156, 899)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 779 567522 613125 823070 436843 462978 991627 806678 903041 542497 547219 596709 745967 399856 145590 965538 241999 760481 360456 856982 227676 950994 337074 913568 > 8¹⁵⁶ [i]