MinT - Best Known (140−61, 140, s)-Nets in Base 9

Best Known (140−61, 140, s)-Nets in Base 9

(140−61, 140, 344)-Net over F₉ — Constructive and digital

Digital (79, 140, 344)-net over F₉, using

4 times m-reduction [i] based on digital (79, 144, 344)-net over F₉, using
- trace code for nets [i] based on digital (7, 72, 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]

(140−61, 140, 488)-Net over F₉ — Digital

Digital (79, 140, 488)-net over F₉, using

trace code for nets [i] based on digital (9, 70, 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]

(140−61, 140, 39702)-Net in Base 9 — Upper bound on s

There is no (79, 140, 39703)-net in base 9, because

1 times m-reduction [i] would yield (79, 139, 39703)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 4 362404 473113 432571 627080 008412 155645 615587 129801 786685 615766 375208 178085 591897 242792 931034 581485 139156 287568 920344 113888 768427 984017 > 9¹³⁹ [i]