MinT - Best Known (137−65, 137, s)-Nets in Base 9

Best Known (137−65, 137, s)-Nets in Base 9

(137−65, 137, 232)-Net over F₉ — Constructive and digital

Digital (72, 137, 232)-net over F₉, using

3 times m-reduction [i] based on digital (72, 140, 232)-net over F₉, using
- trace code for nets [i] based on digital (2, 70, 116)-net over F₈₁, using
  - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
      - a function field by SÃ©mirat [i]

(137−65, 137, 321)-Net over F₉ — Digital

Digital (72, 137, 321)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(137−65, 137, 18149)-Net in Base 9 — Upper bound on s

There is no (72, 137, 18150)-net in base 9, because

1 times m-reduction [i] would yield (72, 136, 18150)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 5993 470180 784820 780939 025893 664999 727967 431165 249003 450558 906832 453697 803603 842737 129616 243511 751115 206134 049548 749921 255113 445889 > 9¹³⁶ [i]