MinT - Best Known (137−129, 137, s)-Nets in Base 9

Best Known (137−129, 137, s)-Nets in Base 9

(137−129, 137, 40)-Net over F₉ — Constructive and digital

Digital (8, 137, 40)-net over F₉, using

net from sequence [i] based on digital (8, 39)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 8 and N(F) ≥ 40, using
  - a function field by SÃ©mirat [i]

(137−129, 137, 42)-Net over F₉ — Digital

Digital (8, 137, 42)-net over F₉, using

net from sequence [i] based on digital (8, 41)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 8 and N(F) ≥ 42, using
  - a curve from the manYPoints database [i]

(137−129, 137, 81)-Net in Base 9 — Upper bound on s

There is no (8, 137, 82)-net in base 9, because

65 times m-reduction [i] would yield (8, 72, 82)-net in base 9, but
- extracting embedded orthogonal array [i] would yield OA(9⁷², 82, S₉, 64), but
  - the linear programming bound shows that MÂ â‰¥ 81 263653 342333 764706 263468 532084 739047 417804 545994 203521 123805 237703 536792 532419 / 153444 711875 > 9⁷² [i]