MinT - Best Known (137−61, 137, s)-Nets in Base 9

Best Known (137−61, 137, s)-Nets in Base 9

(137−61, 137, 344)-Net over F₉ — Constructive and digital

Digital (76, 137, 344)-net over F₉, using

1 times m-reduction [i] based on digital (76, 138, 344)-net over F₉, using
- trace code for nets [i] based on digital (7, 69, 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]

(137−61, 137, 424)-Net over F₉ — Digital

Digital (76, 137, 424)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(137−61, 137, 31867)-Net in Base 9 — Upper bound on s

There is no (76, 137, 31868)-net in base 9, because

1 times m-reduction [i] would yield (76, 136, 31868)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 5986 068404 781121 644170 980688 662976 671804 461936 524416 320239 321184 135618 123711 909938 620892 755336 257720 517979 600952 800328 664721 612993 > 9¹³⁶ [i]