MinT - Best Known (137−57, 137, s)-Nets in Base 9

Best Known (137−57, 137, s)-Nets in Base 9

(137−57, 137, 344)-Net over F₉ — Constructive and digital

Digital (80, 137, 344)-net over F₉, using

9 times m-reduction [i] based on digital (80, 146, 344)-net over F₉, using
- trace code for nets [i] based on digital (7, 73, 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]

(137−57, 137, 583)-Net over F₉ — Digital

Digital (80, 137, 583)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(137−57, 137, 60909)-Net in Base 9 — Upper bound on s

There is no (80, 137, 60910)-net in base 9, because

1 times m-reduction [i] would yield (80, 136, 60910)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 5984 843154 823491 545553 831857 887439 952254 217911 279045 970532 880261 792808 649255 584565 692017 366064 427632 680008 038601 934885 722496 920257 > 9¹³⁶ [i]