MinT - Best Known (127−87, 127, s)-Nets in Base 8

Best Known (127−87, 127, s)-Nets in Base 8

(127−87, 127, 98)-Net over F₈ — Constructive and digital

Digital (40, 127, 98)-net over F₈, using

t-expansion [i] based on digital (37, 127, 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]

(127−87, 127, 129)-Net over F₈ — Digital

Digital (40, 127, 129)-net over F₈, using

t-expansion [i] based on digital (38, 127, 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]

(127−87, 127, 1041)-Net in Base 8 — Upper bound on s

There is no (40, 127, 1042)-net in base 8, because

1 times m-reduction [i] would yield (40, 126, 1042)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 635297 157805 955707 930086 152717 551059 980185 160680 245778 315206 526158 794891 793686 399026 509806 756325 414133 252347 928320 > 8¹²⁶ [i]