MinT - Best Known (136−59, 136, s)-Nets in Base 9

Best Known (136−59, 136, s)-Nets in Base 9

(136−59, 136, 344)-Net over F₉ — Constructive and digital

Digital (77, 136, 344)-net over F₉, using

4 times m-reduction [i] based on digital (77, 140, 344)-net over F₉, using
- trace code for nets [i] based on digital (7, 70, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(136−59, 136, 488)-Net over F₉ — Digital

Digital (77, 136, 488)-net over F₉, using

trace code for nets [i] based on digital (9, 68, 244)-net over F₈₁, using
- net from sequence [i] based on digital (9, 243)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 9 and N(F) ≥ 244, using
    - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₈₁ [i]

(136−59, 136, 40365)-Net in Base 9 — Upper bound on s

There is no (77, 136, 40366)-net in base 9, because

1 times m-reduction [i] would yield (77, 135, 40366)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 665 309620 475985 366007 069406 827602 912063 177798 879973 154662 533780 494216 694403 594826 984553 943571 361260 821489 775933 007527 686478 143281 > 9¹³⁵ [i]