MinT - Best Known (138−128, 138, s)-Nets in Base 9

Best Known (138−128, 138, s)-Nets in Base 9

(138−128, 138, 40)-Net over F₉ — Constructive and digital

Digital (10, 138, 40)-net over F₉, using

t-expansion [i] based on digital (8, 138, 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]

(138−128, 138, 54)-Net over F₉ — Digital

Digital (10, 138, 54)-net over F₉, using

net from sequence [i] based on digital (10, 53)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 10 and N(F) ≥ 54, using
  - a curve from the manYPoints database [i]

(138−128, 138, 96)-Net in Base 9 — Upper bound on s

There is no (10, 138, 97)-net in base 9, because

51 times m-reduction [i] would yield (10, 87, 97)-net in base 9, but
- extracting embedded orthogonal array [i] would yield OA(9⁸⁷, 97, S₉, 77), but
  - the linear programming bound shows that MÂ â‰¥ 577 508372 587460 541806 827932 218426 944109 871880 135970 034666 716258 074835 618754 104688 984869 879637 / 5272 197931 > 9⁸⁷ [i]