MinT - Best Known (138−113, 138, s)-Nets in Base 9

Best Known (138−113, 138, s)-Nets in Base 9

(138−113, 138, 78)-Net over F₉ — Constructive and digital

Digital (25, 138, 78)-net over F₉, using

t-expansion [i] based on digital (22, 138, 78)-net over F₉, using
- net from sequence [i] based on digital (22, 77)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
    - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(138−113, 138, 96)-Net over F₉ — Digital

Digital (25, 138, 96)-net over F₉, using

net from sequence [i] based on digital (25, 95)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 25 and N(F) ≥ 96, using
  - a curve from the manYPoints database [i]

(138−113, 138, 552)-Net in Base 9 — Upper bound on s

There is no (25, 138, 553)-net in base 9, because

1 times m-reduction [i] would yield (25, 137, 553)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 57765 462107 177390 650898 845927 711231 181920 187033 160547 553605 398830 542780 756331 197727 097217 599589 816847 043849 432920 050533 076539 856065 > 9¹³⁷ [i]