MinT - Best Known (86−45, 86, s)-Nets in Base 8

Best Known (86−45, 86, s)-Nets in Base 8

(86−45, 86, 98)-Net over F₈ — Constructive and digital

Digital (41, 86, 98)-net over F₈, using

t-expansion [i] based on digital (37, 86, 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]

(86−45, 86, 130)-Net over F₈ — Digital

Digital (41, 86, 130)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(8⁸⁶, 130, F₈, 3, 45) (dual of [(130, 3), 304, 46]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(8⁸³, 129, F₈, 3, 45) (dual of [(129, 3), 304, 46]-NRT-code), using
  - extended algebraic-geometric NRT-code AG_e(3;F,341P) [i] based on function field F/F₈ with g(F) = 38 and N(F) ≥ 129, using
    - a curve from the manYPoints database [i]

(86−45, 86, 3976)-Net in Base 8 — Upper bound on s

There is no (41, 86, 3977)-net in base 8, because

1 times m-reduction [i] would yield (41, 85, 3977)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 58116 386821 833136 600815 273595 728003 317703 500604 271049 493666 629860 243570 974816 > 8⁸⁵ [i]